I found an old RC car - the kind of thing you get for £5 from Argos - lying in my cupboard, mostly untouched since I got it as a birthday present. Still, it has wheels and motors, what more could you ask for?
I proceeded to rip out all its electronics and battery contacts and cover etc, leaving just the underchasis and two motors. I then put my own little 900mAh lipo in the battery compartment along with a small motor driver board (two h-bridges), with all the wires running through the old contact holes. An STM32 dev board went on top and connects to a Sharp IR rangefinder and xbee radio.
All I had it do to start with was drive forwards, then when the ADC detected the voltage from the Sharp sensor went over the limit that indicated an obstacle was ahead, an interrupt fired which caused the car to reverse back and to the left slightly. This was enough to avoid most collisions so the car could pretty much drive about as it wanted. Later I added remote control from a computer over the xbee, with the same interrupt code for collisions. I was planning to put a GPS and some other sensors on the car, but shortly after making it I ordered all the parts for my upcoming quadcopter, which is going to take the limelight for now.