1 Plot, 2 Scale/Axis

As I said, it's pretty easy to adapt the TwoAxisPlot[] function given in the OP. I'll give two flavors here, named TwoAxisListPlot[] and TwoAxisListLinePlot[]:

TwoAxisListPlot[{f_, g_}] := 
 Module[{fgraph, ggraph, frange, grange, fticks, 
   gticks}, {fgraph, ggraph} = 
   MapIndexed[
    ListPlot[#, Axes -> True, 
      PlotStyle -> ColorData[1][#2[[1]]]] &, {f, g}]; {frange, 
    grange} = 
   Last[PlotRange /. AbsoluteOptions[#, PlotRange]] & /@ {fgraph, 
     ggraph}; 
  fticks = Last[
     Ticks /. 
      AbsoluteOptions[fgraph, 
       Ticks]] /. _RGBColor | _GrayLevel | _Hue :> ColorData[1][1];
  gticks = (MapAt[Function[r, Rescale[r, grange, frange]], #, {1}] & /@
       Last[Ticks /. 
        AbsoluteOptions[ggraph, 
         Ticks]]) /. _RGBColor | _GrayLevel | _Hue -> 
     ColorData[1][2];
  Show[fgraph, 
   ggraph /. 
    Graphics[graph_, s___] :> 
     Graphics[
      GeometricTransformation[graph, 
       RescalingTransform[{{0, 1}, grange}, {{0, 1}, frange}]], s], 
   Axes -> False, Frame -> True, 
   FrameStyle -> {ColorData[1] /@ {1, 2}, {Automatic, Transparent}}, 
   FrameTicks -> {{fticks, gticks}, {Automatic, Automatic}}]]

TwoAxisListLinePlot[{f_, g_}] := 
 Module[{fgraph, ggraph, frange, grange, fticks, 
   gticks}, {fgraph, ggraph} = 
   MapIndexed[
    ListLinePlot[#, Axes -> True, 
      PlotStyle -> ColorData[1][#2[[1]]]] &, {f, g}]; {frange, 
    grange} = 
   Last[PlotRange /. AbsoluteOptions[#, PlotRange]] & /@ {fgraph, 
     ggraph}; 
  fticks = Last[
     Ticks /. 
      AbsoluteOptions[fgraph, 
       Ticks]] /. _RGBColor | _GrayLevel | _Hue :> ColorData[1][1];
  gticks = (MapAt[Function[r, Rescale[r, grange, frange]], #, {1}] & /@
       Last[Ticks /. 
        AbsoluteOptions[ggraph, 
         Ticks]]) /. _RGBColor | _GrayLevel | _Hue -> 
     ColorData[1][2];
  Show[fgraph, 
   ggraph /. 
    Graphics[graph_, s___] :> 
     Graphics[
      GeometricTransformation[graph, 
       RescalingTransform[{{0, 1}, grange}, {{0, 1}, frange}]], s], 
   Axes -> False, Frame -> True, 
   FrameStyle -> {ColorData[1] /@ {1, 2}, {Automatic, Transparent}}, 
   FrameTicks -> {{fticks, gticks}, {Automatic, Automatic}}]]

Test:

d1 = Accumulate[RandomReal[{0, 1}, {100}]];
d2 = Accumulate[RandomReal[{0, 50}, {100}]];
GraphicsGrid[{{ListLinePlot[d1], ListPlot[d2]},
             {TwoAxisListPlot[{d1, d2}], TwoAxisListLinePlot[{d1, d2}]}}]

If you want to use David Park's Presentations package, you can reset the ticks and it will look like this:

data1 = {{1, 1.1}, {2, 1.5}, {3, 0.9}, {4, 2.3}, {5, 1.1}};
data2 = {{1, 1001.1}, {2, 1001.5}, {3, 1000.9}, {4, 1002.3}, {5, 1001.1}};

<<Presentations`
crop[x_] := (x - 1000)
Draw2D[
 {
  {Red, Thickness[0.02], Opacity[0.3], 
   ListDraw[data1, Joined -> True]},
  Blue, ListDraw[{#1, crop[#2]} & @@@ data2, Joined -> True]
  },
 AspectRatio -> 1/GoldenRatio,
 Frame -> True,
 FrameTicks -> {{Automatic,
      CustomTicks[crop, {1001, 1002.2, 0.2, 5},
         CTNumberFunction -> (Style[#, FontColor -> Blue] &)]},
   {Automatic, Automatic}},
 PlotLabel -> Row[{Style["data1", Red], ", ", Style["data2", Blue]}],
 PlotRange -> All
 ]

Even though this question has been flagged as answered, I think the answers are more complicated than they need to be (with respect to the authors). I offer the following, which takes advantage of FrameTics:

(*create 2 lists*)
x1 = Accumulate[RandomVariate[NormalDistribution[0, 1], 100]];
x2 = 25 Accumulate[RandomVariate[NormalDistribution[0, 1], 100]];

(*set nice plot options*)
SetOptions[ListPlot, ImageSize -> 500, Frame -> True, Joined -> True, 
GridLines -> Automatic, PlotStyle -> {{Thick, Red}, {Thick, Blue}}, 
PlotRange -> {{0, 100}, {-50, 50}}, 
FrameLabel -> {"Progress", "Red Line","Descriptive Title", "Blue line"}, 
 LabelStyle -> {12, FontFamily -> "Arial"}];

(*display, using FrameTicks for the bottom axis to show what it does*)
ListPlot[{x1, x2},FrameTicks -> {{{0, "Beginning"}, {25, "Early"},
{50, "Middle"},{75,"AlmostnFinished"}, {100, "Finished"}}, Automatic, None,Automatic}]

As expected, x2 goes off ListPlot's range and needs a different scale. This can be accomplished by rescaleing x2 and using FrameTics to create a rescaled axis on the right. First, rescale x2 using the function rescaled[]:

datamax = Max[x2]; datamin = Min[x2];
datarange = datamax - datamin;
plotrange = 100; plotmin = -50;
rescaled[x_] := (x - datamin) plotrange/datarange + plotmin

Next, create new axis lables for the right axis::

axeslabel[v_] := {rescaled[v], ToString[v]}
rightaxis = Table[axeslabel[v], {v, -500, 500, 100}]

Finally, create the new ListPlot:

lp = ListPlot[{x1, x3},FrameTicks -> {{{0, "Beginning"}, {25, "Early"}, {50, 
  "Middle"}, {75, "AlmostnFinished"}, {100, "Finished"}}, Automatic, None, rightaxis}]
x3 = rescaled[#] & /@ x2;

See how easy that was!

Upon reflection my approach isn't too different from Peter Breitfeld's, except perhaps that I made a more general rescaling routine.

I needed a easy to modify "TwoAxisDateListPlot".

Thanks ArgentoSapiens for the inspiration. Here is my version.

list1 = FinancialData["GE", "Feb. 5, 2014"];
list2 = FinancialData["Gold", "March. 5, 2014"];
TwoAxisDateListPlot3[list1, list2, AspectRatio -> 0.3, ImageSize -> Large]

ClearAll[TwoAxisDateListPlot3]
Needs["Calendar`"];
TwoAxisDateListPlot3[dat1__, dat2__, opts : OptionsPattern[]] := Block[
{data1 = dat1, data2 = dat2, plot1, plot2, userOptions,defaultOptions, minx, maxx, temp},
(* display two datelist-graphs on one diagram *)

(* span x *)
If[DateQ[data1[[1, 1]]] (* find out date format *),
temp = SortBy[data1[[;; , 1]]~Join~data2[[;; , 1]], AbsoluteTime];
minx = temp[[1]]; maxx = temp[[-1]],
minx = Min[{data1[[;; , 1]], data2[[;; , 1]]}]; 
maxx = Max[{data1[[;; , 1]], data2[[;; , 1]]}]
];

(* get options *)
userOptions = FilterRules[{opts}, Options[DateListPlot]];
defaultOptions = FilterRules[{PlotRange -> {{minx, maxx}, {All, All}}, 
ImagePadding -> {{40, 40}, {25, 5}}}, Options[DateListPlot]];

(* do the plots *)
plot1 = DateListPlot[data1, PlotStyle -> Blue, 
  Frame -> {{True, False}, {True, True}},
  FrameStyle -> {Directive[FontFamily -> "Helvetica", Bold], 
  Directive[FontFamily -> "Helvetica", Bold, Blue], Automatic, 
  Automatic}, userOptions, defaultOptions] // Quiet;
plot2 = DateListPlot[data2, PlotStyle -> Darker[Green], Axes -> False, 
  Frame -> {{False, True}, {False, False}},
  FrameTicks -> All,
  FrameStyle -> {Automatic, Automatic, Automatic, 
    Directive[FontFamily -> "Helvetica", Bold, Darker[Green]]}, 
  userOptions, defaultOptions] // Quiet;
Overlay[{plot1, plot2}]];

Most compatible solution:

The solutions provided are not compatible with plots that contain labels. Here's a solution with possibility to add options:

TwoAxisListPlot[{f_, g_}, opts___] := 
 Module[{fgraph, ggraph, frange, grange, fticks, 
   gticks}, {fgraph, ggraph} = 
   MapIndexed[
    ListPlot[#, Axes -> True, PlotStyle -> ColorData[1][#2[[1]]], 
      opts] &, {f, g}]; {frange, 
    grange} = (PlotRange /. 
        AbsoluteOptions[#, PlotRange])[[2]] & /@ {fgraph, ggraph}; 
  fticks = N@FindDivisions[frange, 5];
  gticks = 
   Quiet@Transpose@{fticks, 
      ToString[NumberForm[#, 2], StandardForm] & /@ 
       Rescale[fticks, frange, grange]};
  Show[fgraph, 
   ggraph /. 
    Graphics[graph_, s___] :> 
     Graphics[
      GeometricTransformation[graph, 
       RescalingTransform[{{0, 1}, grange}, {{0, 1}, frange}]], s], 
   Axes -> False, Frame -> True, 
   FrameStyle -> {ColorData[1] /@ {1, 2}, {Automatic, Automatic}}, 
   FrameTicks -> {{fticks, gticks}, {Automatic, Automatic}}]]

And here's how it's called:

TwoAxisListPlot[{Accumulate[RandomReal[{0, 1}, {100}]], 
  Accumulate[RandomReal[{0, 1}, {100}]]}, {Frame -> True, 
  PlotLabel -> "Hello there!", PlotRange -> All, Joined -> True, 
  PlotMarkers -> {Automatic, Small}, ImageSize -> Large, 
  FrameLabel -> {{"Mean magnetic field (T)", 
     "(Hz)"}, {"Some parameter", ""}}, BaseStyle -> {FontSize -> 16}}]

ArgentoSapiens's answer works well, but if the two plots don't have quite the same horizontal range, or you want to add different-sized labels to the two vertical axes, then it can be a bit tricky to line the two plots up correctly in the Overlay. I figured out the following trick that helped a lot with the alignment:

1. Include both plots' frames and labels in each plot. E.g. set Frame -> {{True, True},{True,False}} in both plots, and add the exact same labels to both plots. This way all the spacing will be consistent between the two plots.

2. Set each duplicated feature to be Transparent in one of the two plots (e.g. using FrameStyle). This way the elements will still take up the right amount of space, but when you overlay them, they won't be twice as dark as they should be.

Now when you combine the two plots using Overlay, they should be almost perfectly lined up, and you don't need to worry about setting the ImagePadding. (Although you still may need to tweak the individual plots with ImageSize, and/or the Overlay with the Alignment option, in order to line them up perfectly.)

Also, if you do it this way then the image will be cropped correctly, whereas there will be extra white space around the sides if you set ImagePadding too big.

Here is just a quick update of J.M.'s code to use some newer (read undocumented) functions in the Charting`context.

TwoAxisListPlot[{list1_, list2_}, opts : OptionsPattern[]] := 
 Module[{plot1, plot2, ranges},
  {plot1, plot2} = ListLinePlot /@ {list1, list2};
  ranges = Last@Charting`get2DPlotRange@# & /@ {plot1, plot2};
  ListPlot[
   {list1, Rescale[list2, Last@ranges, First@ranges]},
   Frame -> True,
   FrameTicks -> {{Automatic, 
      Charting`FindTicks[First@ranges, Last@ranges]}, {Automatic, 
      Automatic}},
   FrameStyle -> {{Automatic, ColorData[97][2]}, {Automatic, 
      Automatic}},
   FilterRules[{opts}, Options[ListPlot]]
   ]
  ]

d1 = Accumulate[RandomReal[{0, 1}, {100}]];
d2 = Accumulate[RandomReal[{0, 50}, {100}]];
GraphicsGrid[{{ListLinePlot[d1], 
   ListPlot[d2]}, {TwoAxisListPlot[{d1, d2}], 
   TwoAxisListPlot[{d1, d2}, Joined -> True]}}]

By the upper Mr. Jason B.'s nice codes on TwoAxisListPlot, I made one on DateListPlot as TwoAxisDateListPlot as following, some notations listed for helping others to change it to other ***Plot cases.

TwoAxisDateListPlot[{datelistLeft_?ListQ, dateListRight_?ListQ},opts : OptionsPattern[]] :=
    Module[ {shapedListRight, plotLeft, plotRight, twoRanges, result},

        (*check lists' depth*)
        If[ Or[ArrayDepth@datelistLeft != 2, ArrayDepth@dateListRight != 2],
            Return[$Failed]
        ];

        (* to be reshaped*)
        shapedListRight = dateListRight;

        (*find plots'Ranges*)
        {plotLeft, plotRight} = 
         DateListPlot /@ {datelistLeft, shapedListRight};
        twoRanges = 
         Last@Charting`get2DPlotRange@# & /@ {plotLeft, plotRight};

        (*reshape dataRight*)
        shapedListRight[[;; , 2]] = 
         Rescale[shapedListRight[[;; , 2]], Last@twoRanges, First@twoRanges];

        (*draw together*)
        result = DateListPlot[{datelistLeft, shapedListRight},
          Frame -> True, 
          FrameTicks -> {{Automatic, 
             Charting`FindTicks[First@twoRanges, 
              Last@twoRanges]}, {Automatic, Automatic}}, 
          FrameStyle -> {{ColorData[97][1], ColorData[97][2]}, {Automatic, 
             Automatic}}, FilterRules[{opts}, Options[DateListPlot]]];

        (*return shaped plots*)
        Return[result];
    ]
TwoAxisDateListPlot[datelistLeft_?ListQ, dateListRight_?ListQ, opts : OptionsPattern[]] := TwoAxisDateListPlot[{datelistLeft, dateListRight}, opts]

simple demos as :

list1 = FinancialData["NYSE:IBM", "March. 8, 2015"];
list2 = FinancialData["NASDAQ:AAPL", "March. 8, 2015"];

TwoAxisDateListPlot[list1, list2]

If some options given, it shows as this.

TwoAxisDateListPlot[list1, list2,
 DateTicksFormat -> {"MonthNameShort", ".", "Year"},
 FrameLabel -> {{Style["IBM", Larger, Bold], 
    Style["AAPL", Larger, Bold]}, {None, None}},
 PlotLabel -> Style["Stocks Comparsion", 18, Bold]
 ]

If PlotTheme be used, a new version comes up

 TwoAxisDateListPlot[{datelistLeft_?ListQ, dateListRight_?ListQ}, opts : OptionsPattern[]] :=
        Module[ {shapedListRight, plotLeft, plotRight, twoRanges, theme, colors, result},

           (*check lists' depth*)
            If[ Or[ArrayDepth@datelistLeft != 2, ArrayDepth@dateListRight != 2],
                Return[$Failed]
            ];

            (*get colors*)
            theme = Lookup[Association[opts], PlotTheme, Automatic];
            colors = 
             Most /@ PadRight[{}, {2}, 
               "DefaultPlotStyle" /. (Method /. 
                  Charting`ResolvePlotTheme[theme, DateListPlot])];

            (* to be reshaped*)
            shapedListRight = dateListRight;

            (*find plots'Ranges*)
            {plotLeft, plotRight} = 
             DateListPlot /@ {datelistLeft, shapedListRight};
            twoRanges = 
             Last@Charting`get2DPlotRange@# & /@ {plotLeft, plotRight};

            (*reshape dataRight*)
            shapedListRight[[;; , 2]] = 
             Rescale[shapedListRight[[;; , 2]], Last@twoRanges, First@twoRanges];

            (*draw together*)
            result = DateListPlot[{datelistLeft, shapedListRight},
              Frame -> True, 
              FrameTicks -> {{Automatic, 
                 Charting`FindTicks[First@twoRanges, 
                  Last@twoRanges]}, {Automatic, Automatic}},
              FrameStyle -> {colors, {Automatic, Automatic}}, 
              FilterRules[{opts}, Options[DateListPlot]]];

            (*return shaped plots*)
            Return[result];
        ]
    TwoAxisDateListPlot[datelistLeft_?ListQ, dateListRight_?ListQ, opts : OptionsPattern[]] := TwoAxisDateListPlot[{datelistLeft, dateListRight}, opts]

demos as

TwoAxisDateListPlot[list1, list2,
 DateTicksFormat -> {"MonthNameShort", ".", "Year"},
 FrameLabel -> {{Style["IBM", Larger, Bold], 
    Style["AAPL", Larger, Bold]}, {None, None}},
 PlotLabel -> Style["Stocks Comparsion", 18, Bold],
 PlotTheme -> "Marketing"
 ]

or

TwoAxisDateListPlot[list1, list2,
 DateTicksFormat -> {"MonthNameShort", ".", "Year"},
 FrameLabel -> {{Style["IBM", Larger, Bold], 
    Style["AAPL", Larger, Bold]}, {None, None}},
 PlotLabel -> Style["Stocks Comparsion", 18, Bold],
 PlotTheme -> "Business"
 ]

or

TwoAxisDateListPlot[list1, list2,
 DateTicksFormat -> {"MonthNameShort", ".", "Year"},
 FrameLabel -> {{Style["IBM", Larger, Bold], 
    Style["AAPL", Larger, Bold]}, {None, None}},
 PlotLabel -> Style["Stocks Comparsion", 18, Bold],
 PlotTheme -> "Detailed"
 ]

There's always one for using.

ClearAll[TwoAxisDateListPlotV12]
Needs["Calendar`"];
TwoAxisDateListPlotV12[dat1__, dat2__, opts : OptionsPattern[]] := 
  Block[{data1 = dat1, data2 = dat2, plot1, plot2, userOptions, 
    defaultOptions, minx, maxx, temp},
   (*display two datelist-graphs on one diagram*)
   (*span x*)
   
   If[DateQ[First@data1["Dates"]] 
     (*find out date format*), 
     temp = SortBy[data1["Dates"]~Join~data2["Dates"], AbsoluteTime];
     minx = First@AbsoluteTime[#] & /@ temp["Dates"]; 
     maxx = Last@AbsoluteTime[#] & /@ temp["Dates"] - 1; 
     minx = Min[{AbsoluteTime[#] & /@ data1["Dates"], 
        AbsoluteTime[#] & /@ data2["Dates"]}];
     maxx = 
      Max[{AbsoluteTime[#] & /@ data1["Dates"], 
        AbsoluteTime[#] & /@ data2["Dates"]}];]
    (*get options*)
    
    userOptions = FilterRules[{opts}, Options[DateListPlot]];
   defaultOptions = 
    FilterRules[{PlotRange -> {{minx, maxx}, {All, All}}, 
      ImagePadding -> {{40, 40}, {25, 5}}}, Options[DateListPlot]];
   (*do the plots*)
   plot1 = DateListPlot[data1, PlotStyle -> Blue, 
      Frame -> {{True, False}, {True, False}}, 
      FrameStyle -> {Directive[FontFamily -> "Helvetica", Bold], 
        Directive[FontFamily -> "Helvetica", Bold, Blue], Automatic, 
        Automatic}, userOptions, defaultOptions] // Quiet;
   plot2 = 
    DateListPlot[data2, PlotStyle -> Darker[Green], Axes -> False, 
      Frame -> {{False, True}, {False, False}}, FrameTicks -> All, 
      FrameStyle -> {Automatic, Automatic, Automatic, 
        Directive[FontFamily -> "Helvetica", Bold, Darker[Green]]}, 
      userOptions, defaultOptions] // Quiet;
   Overlay[{plot1, plot2}]];

---

list1 = FinancialData["APPL", "Febuary 5, 2015"];
list2 = FinancialData["GOOGL", "March 5, 2015"];
TwoAxisDateListPlotV12[list1, list2, AspectRatio -> 0.3, 
 ImageSize -> Large]

Using ResourceFunction["CombinePlots"], you can combine arbitrary plots into two-axes plots, without the need to create a new version for every type of plot:

ResourceFunction["CombinePlots"][
 ListLinePlot[
  Accumulate[RandomReal[{0, 1}, {100}]],
  PlotStyle -> Blue, Frame -> True, FrameStyle -> Blue
  ],
 Plot[
  x^2, {x, 10, 100},
  PlotStyle -> Red, Frame -> True, FrameStyle -> Red
  ],
 "AxesSides" -> "TwoY"
 ]

Note that the frame styles of the two plots where automatically applied to the appropriate axes, and that it is easy to combine Plot together with ListPlot. (CombinePlots also works with different ScalingFunctions etc, see the documentation for more details & examples)