First of all, good choice on using the Arduino UNO! 
As for the robot, you can build your own motor control circuit using a transistor (probably a MOSFET), or a motor driver chip, but I would go with a motor driver shield because of its simplicity and ease of use. If you want one for really cheap, check on ebay for a motor driver shield. An even better option would be to use servos. You don't even need to use a motor control board for those! You can get them on ebay for cheap as well. Make sure to get continuous rotation, because if they aren't, they might only be able to turn a short range. 
As a final word, if you choose to go with the motor shield, then use the Adafruit one. I love SparkFun, but I just have to go with Adafruit on this one. The servos would still be the best and most controllable (precise) option.
Hope your project goes well!
~Dylan