Do I need to use two wire types for an indoor/outdoor run to a garage?

Put it in conduit, use THWN, done. Most THHN is also marked THWN or THWN-2 (and MTW, and perhaps a few other listings.)

Because part of the conduit is outside, the wire used must be wet rated (W) but there's no need to change wire type where it goes inside the building.

Using 6/3 cable in conduit is a right royal pain in the behind you don't need or want. Use individual wires when using conduit - they pull easier and they typically allow you to use smaller conduit per the way the fill rules treat cables .vs. wires.

If you use continuous metallic conduit, that is your ground, you don't need a grounding wire, and you can use 3/4" conduit (minimum.) If you use PVC conduit you need a grounding wire and you'll need 1 inch schedule 80 minimum (which most "home centers" will not stock, but any electrical supply will have tons of) because it sounds like most of the run is going to be below 8 feet and thus considered "subject to damage" so it needs the additional protection of the heavier wall of schedule 80. Don't use Sch 40 becasue it's all you can find at the home center. Go find the electrical supply. The wire may cost less there too...