A pair of folding grocery panniers can haul home a week's worth of produce and meat for a family of four, or a month's worth of staples. Less if you eat a lot of pre-packaged stuff like frozen dinners or drink a lot of soda/milk, but that would just mean you need to factor in an extra milk run. You can hang 'em off the rear rack of pretty much any bike you can attach a rear rack to, just make sure the rack is sturdy. (Not expensive, just sturdy.)

As to what kind of bike, it depends on:

1) What's the ride like? Flat and easy? Lots of hills? Rough pavement or dirt road/trails?
2) How in shape are you? Do you like to ride hard and fast, or do you like to cruise and ride to relax?
3) What other kinds of riding would you like to do? Bike paths, mountain biking, road racing, long range touring?