It all depends on how you’re driving your wheels. If youre driving them with the bolt on sprocket style, the wheel just rests on the shaft and has its own bearings. A 3/8" steel shaft works fine for this, even 5/16" has been used with no problems.
However if you are using a shaft driven wheel, I would definitely use a 1/2" steel shaft with a 1/8" keyway. Just for security. If you want to use aluminum, then you should use a 5/8" shaft with a 1/8" keyway. Of course there are always variations. If you have a small 3" wheel, you can easily run it on a 3/8" shaft because it isnt taking so much torque, especially if the shaft is supported on both sides of the wheel.
Just use your common sense and ask yourself, what forces is this taking, and how big? If The wheel is run by bolt on sprocket, then the shaft only takes weight. If the wheel is shaft driven, then it takes torque and weight, so its gotta be stronger. If the wheel is smaller (high speed, low torque) then the shaft can be smaller than if the wheel were bigger (low speed, high torque). Just take all these things into account, and figure out the best shaft size.
A few notes: keyways are superior to set screws and roll pins in transmitting torque. I had to learn this the hard way freshman year. Also, the plastic skyway wheels can be stronger than the shaft they run on. You can break the shaft without breaking the wheel.
As for transmissions, it all depends on speed again. Your motors can only give a maximum power rating, so a high RPM shaft will take less torque than an low RPM shaft. Thus, my own transmissions have always had high RPM outputs, just to keep the shafts small (lowest RPM I’ve geared to was 1000 rpm at no load, on two CIMs). If you keep relatively high gearbox shaft speeds, you can easily have your entire transmission running 3/8" or 5/16" shafts. For shifters, I have always used 1/2" shafts on the shifter itself, for safety, since shifting can be rough.
As for your bearing question, I’ve always used steel bearings on everything. I dont know what type of bearings you have, whether they are thermoplastic or ceramic based plastic will change their load and rpm ratings significantly. It’s best if you can get the bearings specs, just to be sure.
Hope that helps.