Shifting sucks with my training wheels compared to my race wheels
Whenever I have my racing wheels in, my bike shifts fine. But when I have my training wheels on, the derailleur and the cassette seems to out of line or something. It jumps everywhere and it shifts horrible. It's an 8 speed bike with both wheels with 8 speed cassettes. The training wheel has an older cassette on. Could that be the problem? But it doesn't seem like a cassette problem b/c when I have it on the highest gear on my training wheel, the chain goes beyond where the teeth are and falls off the cassette. I have no idea what's wrong with it.
It's likely that the cassette position on the hub is slightly different on each set of wheels, which means the derailleur isn't quite lining up properly. You'll probably need to adjust the limit screws on the derailleur when you change wheels to fix the alignment issues. Your bike sounds like it has been aligned for the racing wheels. The derailleur forcing the chain off to the outside of the small cog would indicate that the training wheel cassette is positioned slightly further in (toward the hub).
Check out Park Tool repair help site for instructions on making the adjustment.