If you accept that more weather extremes are likely, then moving to setting up the water system in a way that it's less likely to actually freeze, as we do in the (traditional) colder areas might look worthwhile. However, to your question:

Your basic approach, for a more limited set of changes, in advance, make sure that you can (and know how to, and preferably have documented for the rest of your family, or the next owner of your house) turn off your water supply & drain your pipes as fully as possible. This might involve adding some drain valves at low points, if the system is not currently set up for that.

In the event, you should turn off your water supply and drain the pipes, the water heater, the water softener, etc..., and put RV anti-freeze (which you will have stockpiled in advance for the event, rather than try to go buy while the shelves are empty - or salt solution if you miss that point) into the toilets and other traps (maintaining the trap seal while preventing damage from freezing.)

That should isolate any freeze damage to the water supply system side of the supply valve. If you are on a well, you may have more area you are personally responsible for (and you would turn off the well pump power supply, rather than shutting off the supply valve.)

Use the Minnesota and California building codes

That's to protect you from freeze and earthquakes. (New Madrid fault, yo).

There are a thousand little details in home construction that differ between warm places and cold. I come from the snow belt, and am constantly looking at sun-belt houses going "Wow, I guess you can do that here". Bare water main popping out of the ground and going into the house, "sure, why not". In the midwest that's buried below the frost-line, all the way into climate controlled space (e.g. basement).

Look at how they design buildings, how they avoid e.g. putting water pipes on exterior walls, etc. Use that as a guide when doing renovations.

If you see any "low hanging fruit" opportunities to move a pipe from the outside of the house's insulation envelope to the inside... take it.

If you have areas you just can't protect, I'd advise a combination of hyperinsulation + thermal mass, and "heat tape". Electric heat tape actively warms the pipes (when electricity is available).

Hyper-insulation of pipes slows the inevitable. Let's say you do a great job of insulating the pipe, and at 5 degrees F, it loses 1 BTU per hour. (1 BTU = energy to make 1 pound of water rise or fall 1 degree F.) The problem is, the pipe is small and doesn't contain much water. If it has a half pound, and started at 50F (18 degrees to freezing), it only has 9 BTUs to lose before it starts to freeze. My answer is to greatly increase the thermal mass that is inside the hyperinsulation wrappings, by adding water volume to it. 2-litre soda bottles will suffice - cheap and expendable. Hyper-insulate all that - the surface area is much larger, so it will lose, say 5 BTUs/hour. But now if it has 20 pounds of water instead of 0.5 pounds, it now has 180 BTUs to give up! That's more like it.

That, plus heat tape is a winning combination, because it means the pipe (and bottles) have as many BTUs as possible to start with, and it only has to make it across the lesser of the power outage and the cold snap.

Have a generator interlock, for Pete's sake

This is a simple "sliding plate" affair that prevents both the main and an installed breaker from being switched on. Nothing more than that.

Generators (or much better, electric cars with inverters such as the 2004 Silverado Hybrid) are now so prolific that almost anyone could be in a situation of having power out and also having a generator. Without the necessary "bit" to connect them, tragedy happens, to the point where major-storm events now accrue most of their fatalities from generator misuse. I hate generators with the heat of 1000 suns, but I feel every house should have an interlock that lets them cut over either the entire panel, or a critical-loads subpanel, over to generator because this is better than dying.

A lot of the deaths aren't Darwin awards. A lot of people did stuff that seemed perfectly reasonable, and were blindsided by circumstances.

Note that a $350 "transfer switch" is a weak way to do this - it's expensive and supports only 6, 8 or 10 circuits. It should not be used, unless dealing with an extremely archaic panel for which an interlock is simply impossible. But even then, an $80 "critical loads" subpanel is a better way to do this, and far cheaper as well.

Most panels support a simple $25-$80 sliding-plate interlock between the main breaker and one other breaker ($10). That breaker is then wired to a generator "inlet" in an appropriate location. Shut all breakers off; turn the generator backfeed breaker on; then turn on select loads.

Have an auxiliary furnace

One thing that blows my mind is all the houses in the snowbelt with gas heat, which cannot function if electricity is not also present. That's just not a problem in the Sunbelt, as we have the famous Empire style wall or floor furnace - a gas-fired, convection-circulated furnace. They work fine during outages as they do not use electricity at all (except for millivolts which they generate with their own thermocouple; this allows use of an external thermostat).

It helps to design a house around such a furnace, as convection is the name of the game. But even if not, they will add enough heat to the building envelope to at least prevent freeze.

The units are inexpensive - under $1000 for a 50,000 BTU unit. They do require a vent stack, of course, and they do draw negative pressure on the building where that is a concern. There may be types which draw intake air from outside, but being unpowered makes that a design challenge.

Gas/Propane heat + a way to hook to a generator

My in-laws had a severe ice storm a few years ago (no power for 36 hours) and my father-in-law knew how to pull the power line to the HVAC system and run the blower off his generator.

Harper's interlock sounds good, but a cheaper DIY solution (provided you're comfortable with your electrical panel) is doing this a double-pole double-throw switch coupled with an inlet (15A shown but you can find 20A varieties). Your DPDT switch can switch power from the breaker box to your generator (NEVER backfeed your generator into the panel), and the inlet lets you plug a standard extension cord up. Make sure you get enough wattage to run your HVAC blower by itself. In most cases they take 20A or less (my FIL ran his on an 1800W invetor generator and most truck generators are in the 2000W range). This setup is simple enough for anyone to use.

Why not an interlock? Aside from the cost of the interlock and likely an electrician, you might not want to try to power your whole house off one generator. The best you can do is 30A 240v off one outlet. It will run your lights, TV and most electronics (provided it's an inverter generator). You can even squeeze an appliance or two in (my fridge is 3A). But if you're running all that and your furnace blower kicks in (or worse, electric heat), you might be over budget. Extension cords are a better way to hook up individual devices as you need them.

If you really want to run your whole house, just buy a whole-house generator. Not only do they provide more than enough power, they typically include the interlock (plus they generally start automatically).

Here in Minnesota, it's pretty common that a house needs to be winterized. That was especially true about 10 years ago during the surge in foreclosures that left a lot of homes vacant for many months until the banks could move ahead with selling the property. The other answers hit a lot of the key points, but I'll add a few that will more or less guarantee that the plumbing will survive even a hard freeze.

• Turn off the water at the most upstream valve you are able to access. Ours are usually in the front yard at the bottom of a vertical pipe, and require a special wrench called a "street key" to reach them.
• Open all of the faucets, sillcocks, and drain ports. Turn on all tubs and showers.
• Switch the water softener to bypass mode and follow manufacturer instructions to relieve the pressure and drain the tank. Dump out the brine tank and make sure the head is drained.
• Empty the water heater, including any water that might be left in the expansion tank (if you have one). Set the thermostat to "OFF" (Gas) or turn off (and lock out, if possible) the breaker (Electric)
• Disconnect any other appliance that has a water supply line (washing machine, ice maker, fridge, dish washer, etc...). Drain them according to instructions.
• If you have an air compressor, connect it to the system and blow out any water that remains (not strictly needed but if you have a compressor, it's extra insurance)

That takes care of the supply lines, but you should also prepare the waste side of the plumbing system.

• Flush all toilets and hold down the handle until the tank is completely empty (the flapper will generally close well before the tank is fully drained if you don't hold it open).
• Remove traps from under sinks and tubs/showers and empty them out. replace them and stuff rags into the drain to block sewer gas and rats/cockroaches/other bad stuff.
• any drains that you are not able to fully empty should be filled with anti-freeze. This includes any floor drains, clean-outs, and buried drain pipes that might hold water.
• fill toilet bowls with anti-freeze and seal the bowl with shrink wrap to keep them from drying out.

When we had our last big freeze in Ireland, our house was saved by a dripping tap that I had not got around to replacing. There was a constant small flow of water so nothing froze up. We supplied four other houses for a week that winter!