Go on Amazon and look up "Redcat Racing" - ah, what the heck - here's a link:
Check out the videos, check out the comments - it ain't a Traxxas E-Maxx - but you won't be paying that kind of cash, either. It seems like a good deal.
Since it's a standard hobby-grade RC vehicle, it uses a regular ESC (for a brushed motor), and a regular servo for steering - and a regular 2.4 GHz radio system. It comes will all the parts - plus batteries, radio gear, charger, etc - but you can put more money in if you want or need (I would recommend getting another battery or 2, plus a smart charger - and maybe a banana plug (redcat) to tamiya battery adapter - which is more standard - do some research here, so you know what you need to get).
You can then hook the ESC and steering servo up to the Arduino easily, and control them using the Servo library.
There's your basic robot. Add on another servo, plus a bracket, plus an ultrasonic sensor - for a panning sonar system - all of that can still fit on the Arduino. You'll have to figure out a way to mount it to the front of the vehicle - ideally fairly solid.
The car's a beefy beast - fairly fast, fairly tough. Plenty of spare parts and hop-up parts available too.
I know you said that you didn't want to hack on something, or use tools, etc - but you're going to find that if you want to do robotics, you're going to need to get into knowing how to do things like that. It just comes with the territory.
Also - check out Servo City in the meantime, too - if you have the money - they sell a ton of compatible parts, gearing, brackets, and aluminum extrusion and such to all you to easily build robots (again, if you have the cash - the stuff isn't cheap, but it is high quality):
That's a "mars rover" built (partially) with what they sell - Beatty posts here every now and then; he's really dedicated to teaching and instilling in his daughters the importance and knowledge of a STEM education - a great guy and dad, by all accounts - and his kids seem to have taken to it, too!
Lastly - RobotShop is a good place to go to review things - but at times, their stuff seems like it is really marked up or something - I would use them as a reference only, and only buy from them if you can't find it elsewhere cheaper. Many times, you can - if you look.
Oh - one more thing - if you ever do want to "hack" on a junk toy RC car like jremington proposed - check out this thread: