What shield do I need?

Hello,

I currently have a school project involving programming a Adruino with 4 mecanum wheels powerd by an electrical engine. It will need to drive over a black line with 3 sensors looking at the line. The sensors are not the problems but I have some hard time getting to know what I should do to make all the 4 engines movable to give the vehicle its directions. I added a picture of how the engines will move to actually make it move and in what direction.

https://www.google.nl/search?q=mecanum+wheels&safe=off&espv=2&biw=929&bih=747&source=lnms&tbm=isch&sa=X&ei=UUgYVODBDIa1O_OogOAD&ved=0CAYQ_AUoAQ#safe=off&tbm=isch&q=mecanum+wheels+directions&facrc=_&imgdii=_&imgrc=qEtycWusJP2UaM%253A%3B7mr20PDKX5u26M%3Bhttp%253A%252F%252Fwww.robotshop.com%252Fmedia%252Ffiles%252Fimages%252Fdfrobotshop-rover-mecanum-wheel-motion.jpg%3Bhttp%253A%252F%252Fwww.robotshop.com%252Fen%252Fdfrobotshop-rover-20-arduino-mecanum-robot-basic.html%3B500%3B536

My question is do I need a certain shield or what do I need? I picked the mega for the ram it has to make sure it will be able to think fast about dirrections and so on.
I could be off with the thoughts about it since I only had arduino once before and no certain teachers having proper knowledge about them so I would like to get some guidance in picking the right things for the job.

For 4 bi-directional motors you need something like this but I'll leave it to you to verify the voltage and current suitability.