Besides a motor driver, something like this could be used to control the motors forward and reverse but you wouldn't have speed control.
Something like that could be used - but I wouldn't recommend it. Why? Because 12 volt PowerWheel ride-on toys have motors (there are usually 2 motors - one for each rear wheel - the 6 volt toys use a single motor) that can easily draw well over 10 amps of current; you'll quickly fry those small relays on that board. That said, you could use those relays to activate a larger pair of SPDT relays.
First off, I hope you included the 25 amp fuse (I believe it is a slo-blow fuse) when you switched out the battery; if you didn't - add it back in - connect it as close as possible to the battery's positive terminal - a small 6 inch pigtail holder like this is best:
That, or re-use the one that came with the original battery.
Secondly - I would drop using the car battery - especially if it is a standard wet-cell starter battery; while you'll get longer run time initially, in a short amount of time you'll destroy the plates on the battery as it wasn't meant for deep-cycle use. Instead, purchase a larger AGM or gel-cell SLA deep-cycle battery. They are more expensive, but are designed for running loads long-term like this toy.
I would also suggest getting a smart battery charger for the battery (instead of the one the toy came with) - a quality CTEK charger for SLA batteries (or something similar) is best - while more expensive, it will keep your battery in tip-top shape long term (do not use a car charger battery on SLA batteries - the charging current tends to be too high).
Since the fuse is rated at 25 amps, you want a motor driver of at least that much (and actually more), because those motors will pull that much when stalled (that's why the original fuse is rated in such a manner). As I said before, there are two motors, one for each rear wheel. In the PowerWheels toy, the two speeds (high/low) are governed via a clever switch arrangement on the speed "shifter" - the switches basically switch the motors between being connecting in serial vs being connected in parallel across the battery (I can't recall which arrangement is high speed and which is low speed).
I would recommend a couple of other places/items to look into which will be helpful - first, there is a whole community out there of PowerWheels (and similar toys) enthusiasts and modders - they like to do really crazy things with them, mostly as a way to have fun with their kids:
Also google "modified power wheels" for more...
Secondly - one of them has a very nice CD on some of the more popular mods - well worth the cost, as it brings together a ton of information (including patents and service manuals that would take a while to gather yourself):
Ok - with that out of the way - you probably want to control this vehicle, right. Well, as mentioned, you need a motor driver of some sort. What you'll want to find, ultimately, is an h-bridge driver board that can handle about 30 amps. Alternatively, you could connect the motors separately - each will need about a 15 amp driver.
You can find such boards on Ebay, Pololu and other places - just make sure it can handle the amperage of the motors (also note that some of those boards may need a heatsink and/or forced-air fan attached to handle the current - and some don't come that way; you have to add it yourself - just read all the specs carefully).
As noted above, if you just wanted on/off control - you could use relays, but they would need to be able to handle the current. You could either use one of those boards from Ebay as mentioned, and either switch extra relays with them, or possibly install/hook up better relays - or you could build your own relay board. If you go that route, 40-60 amp SPDT automotive relays are fairly cheap and will easily handle the switching of the load. Just hook them up as shown in the relay tutorial for the Arduino and you'll be fine.
Two SPDT relays are hooked up so that the "common" leads of each relay go to each terminal on the motor, and the NC leads of the relays go to the battery positive terminal (via a fuse!) and the NO leads of the relays are connected to the ground (negative) terminal on the battery. Make sure that you use proper gauge wires between the battery, motors and relays (10-12 gauge should be ok), otherwise the wires will heat up, you'll lose current in them, and the motors will run slower.
Another hint/idea: If you set up your drive motors as seperately wired (so you can control the speed and direction of each) - while it is a more complicated setup, it opens the door to doing some potentially fancy things with your PowerWheels (for instance, an electronic differential and/or traction control!).
Steering will be your greatest issue to tackle - these ride-on toys were not designed with ease of conversion in mind. If you want to throw money at it, you can find fairly powerful servos or linear actuators out there that will do the trick (then all you need to do is figure out how to mount and connect the device to the steering system). Check out Servo City (http://www.servocity.com/) as well as Firgelli Automation (http://firgelliauto.com/) for ideas, just be prepared for sticker shock.
Whatever you do, don't go cheap or small on the gearmotor for actuation of the steering; it needs to be pretty damn robust. For my project, I used have a gearmotor from a drill powering a piece of threaded rod with a travelling nut to actuate my steering. Also - whatever you do, make sure that you have limit switches or some other means to cut the current to the steering motor for your testing and actual operation - if not, you'll either break the motor, or the steering system, or both (ask me how I know). A good limit switch circuit (and also shows how to wire the relays) is shown here - this is Zoomcat's circuit; he's a regular here on the Arduino forum's:
Be aware that the diodes, limit switches, and relays need to be sized for the motor current draw and voltage needs.
If you decide to build your own actuator, you'll also probably want some kind of position feedback; if you use a geared servo from Servo City, that will be built in, of course - but if you use a linear actuator, some may have pulse sensors (typically a hall-effect sensor or reed-relay actuated by magnet) or they may have a potentiometer that you can read. Some linear actuators ($$$) can be driven using servo signals, as well (in which case, you don't need to worry about a separate position sensor). Whatever you do, don't rely on the position sensor in lieu of the limit switches! If that sensor fails for some reason, the limit switches will be your backup.
The PowerWheels modifier community also has numerous instances showing how they modified their particular chassis for steering and R/C control - so look through the forums and other sources to get some ideas there as well.
Lastly - whatever you do - don't try to replace the motors with brushless motors of a similar size. While it is possible, the gearboxes aren't built for the higher RPMs and power those motors can generated, and you will end up burning through gearboxes as a result (and they aren't easy to get as replacement components).
Good luck with your project!