JakeTH:
I also took your advice about the RC car and found an old RC Jeep under my desk. I've removed the top half of the chassis, ripped out the circuit board, and modified the battery compartment to support the Arduino Uno. I want to connect the two motors (as far as I can tell, the front motor turns the two front wheels left and right and the back motor rotates the two back wheels) to the Arduino, but I don't really know how.
If you can identify the h-bridge controllers, or the controller IC on the receiver board (in that first picture, I am assuming the board on the right is the receiver?), you might just be able to use that board as the motor controller. Do you have (or can you get) any high-res pictures of that board (or the main IC on it?); if it is an RX2 IC, you might be fully set.
JakeTH:
I assume I'll need to buy a Motor Shield?
Likely an L298-based motor shield will work fine; you will want to find out what the stall current ratings are for each of the motors in your RC car; if they are each rated at 2 amps or lower when stalled (you can measure the stall current with a multimeter), then a single L298 motor shield will suffice, otherwise you will want one shield for each motor (you may have to modify the shield or wire them up in a "non-stacked" fashion, though - depending on how pin selection and pass-thru is set up for the particular shield you use).
JakeTH:
Will I need some adapter cables to connect the RC motor cables to the Shield? What type of adapter cables? Or do I just pull the white plug off the cables and attach the wires directly to the Motor Shield? Is there anything else I should know about using this Jeep with the Uno?
I would personally look for L298-based driver boards that use screw terminals, like these:
...then cut the connectors off and use bare wire (twist the ends and tin them with solder first for best results).