Potatoes are some of the most idiot-proof vegetables you can grow, simply:
1) Get some potatoes
2) Put them in some dirt in the ground
3) Cover them up with soil
4) Leave them for a few months while a potato plant grows
5) Dig 'em up, and you've got more potatoes.
Check out The Vegetable & Herb Expert to get further information on how to do it properly + what varieties to plant.