Deep water culture is the way to go for its ease of setup and operation.
If you're just breaking into it, you'll need a 5 gallon bucket(with a lid), an air pump, some air stones and connection hoses, a net pot (or two) and some aquarium gravel. Other than the net pots, you can buy all that stuff at wally.
The most expensive piece will be your air pump. don't scrimp on that either. wally has a nice dual output for about $20 that I use.
total setup will run under $40 (nutrient solution excluded) and will grow some mighty fine tomatoes and peppers.
Order your nutrients online and have them send a handful of 2" net pots at the same time. (they're dirt cheap)