Thanks Chagrin for the options. I just looked at the encoder, and while I don't understand everything right now, that will probably change in a few weeks once I learn a bit more about these kinds of things.
Angelo - thanks for explaining why the gyro won't work. Much appreciated. As for the turns, my plan is to have the user manually indicate a turn is coming up before the chair actually turns. Just like in a car, you (supposed to anyway) activate the turn signal prior to actually turning the wheel.
Depending on my time and materials, I might just make use of the turn detection to turn OFF the turn signal rather than just using a constant time. That method would be even more like a car with an automatic deactivation of the turn signal, but this will be a fine tuning of stuff that is already there.
Looks like the Pololu encoder can detect forwards and backwards motion as well. Once I learn a bit more, I will revisit the datasheet, as the price is certainly in my budget.
Feel free to keep the ideas coming. I'm likely going to learn more with this project than anything else I do.