If you want a really cheap chassis, go down to your local thrift store (here in the US try Goodwill) and find an old radio-controlled car/truck/tank/etc - real easy to find, and you shouldn’t spend more than $5-10.00 (US).
Before you buy it, make sure everything seems mechanically sound from the outside:
Turn the drive wheels slowly and listen to the gearbox (don’t turn the wheel fast, you can strip the gears) to make sure it sounds good. Rotate the drive wheels fully a few revolutions to ensure that there isn’t any stripping of the gears (the wheel will slip or “freewheel” if there is).
If the vehicle uses a differential drive system for the driving wheels, remember to lock up one side, turn the other, then lock that side, and turn the other - to check all parts.
If the vehicle uses Ackerman steering, check to make sure that the servo or whatever it is using seems like it works (stay away from the real el-cheapo models that use an electromagnet or such to click-clack the wheels back and forth). If it seems too loose or broken, you may want to pass on it (unless you think you can fix it by using a cheap hobby servo).
Don’t worry if it doesn’t come with the remote - you won’t need it.
Once you decide on a device, and bring it home, you’ll need to put on your hacker’s hat. Carefully open it up, and inspect everything again. If you can, clean and re-lubricate the gearboxes and such (use something like SuperLube or in a pinch, petroleum jelly).
Find the wires from the motors and/or servos and trace them back to the control board. On that board, for the drive motors (and perhaps the steering servo/motor), near where the wires enter the board, you should find them connected to a series of small transistors (or if you are lucky, a darlington transistor array like a ULN2803); basically, these transistors will form (in some manner) an H-bridge (unless you get really unlucky and happen upon a drive scheme like they used in the old Milton Bradley Big Trak - it was some weird two transistor half H-bridge scheme that split the battery box in two; probably saved them a ton of money in manufacturing, though); you’ll want to lay out a schematic for what you find.
Once you have a rough idea of the schematic (the inputs and outputs of the bridge, power, and ground), try hooking up some batteries to the vehicle; then attach a separate 5-6 volt DC battery pack in parallel to act as a “signal source”, and touch the positive wire lead (the negative is hooked to the chassis ground) to the input (base lead bias resistor) of the suspected H-bridge; if all goes well, it should spin the motor in one direction.
If you get lucky, no magic smoke will be lost. If you don’t get anything, no matter what you try, either there is an issue with the circuit, or with the motors; try disconnecting the motors from the control PCB (label the leads first so you can re-connect later if needed), and applying power to see if the motors run. You might find it cheaper and easier to build your own h-bridge circuitry (very easy to do) to drive the motors directly and by-pass the control board. If you do get the control board working, you will want to cut the traces leading from the R/C circuitry outputs to the h-bridge inputs to bypass the circuitry (as well as eliminating other potential problems) - these inputs you would want to buffer (using a hex-buffer chip or similar), then drive the buffer with the Arduino’s digital outputs.
If there are steering servos involved, it is likely that (if it isn’t a cheapo setup) that the servo feedback system is “on-board” the R/C control board; I wouldn’t bother trying to figure it out (its possible, but it could be a nightmare, especially if you don’t understand how a servo feedback circuit works) - instead, hook into the h-bridge or build your own like above, then take the feedback potentiometer and run it to one of the analog inputs on the Arduino. As you control the motor for steering, check the analog values (ideally, you would set up an interrupt on these pins, and check it that way), and when they match up to where you want the angle to be, stop the motor. Alternatively, you could replace the system with a hobby servo. Cheap hobby servos can be found here: http://www.hobbycity.com/hobbycity/store/uh_index.asp
All in all, the idea is to repurpose that which has already been built; this kind of system can (and should) be scaled up as you go along (the same techniques can be used to control a PowerWheels, a small quad, or even a full-size vehicle; of course the control system gets a lot more complex the bigger the vehicle involved, not to mention the safety systems).
I hope this post helps you out!