OK, so it must have some kind of variable program. Given a start signal it moves forward for some distance, or time, or snesor input. Then it turns to the right, and continues for some additional distance, or time, or sensor input.
Depending on the amount of motion program storage a complicated program could be developed.
The servo motors used in radio control can be controlled without radios. They can be controlled with an ordinary microcontroller. I suggest you start by investigating those motors. Then I would look into drive motors. If you have any kind of R/C shop in your neighborhood I would spend an afternoon with those guys.
You might find it useful to get an R/C car or two for ideas and for a cheap source of parts.
Good Luck