It's always a fun and interesting process to make your own robot, car or other mobile machine that's controlled by an Arduino - however after some planning the cost can sometimes get out of control. Or you may not be that interested - instead just curious about the possibilities of doing so. In that case you can save money by taking advantage of the plethora of inexpensive remote-control toys available at various retail outlets. The benefit of doing so is that everything is made for you, and generally cheaper than for what you can do it for yourself.
Then it's a relatively simple matter of modifying the remote control to interface with an Arduino and thus be controlled by it. This is exactly what Andrew Chalkley has demonstrated with a ten-dollar RC car and a little investigation. By cracking open the remote control, he learned that the directional controls were simple push buttons, one for each direction - and thus easily connected to Arduino digital output pins. From that point it's simple to write a sketch (or even a library as Andrew has done) to control the machine. And from that point, you're only limited by imagination and requirements.