For black and white, a developer and fixer are still all you need off the shelf in terms of chemicals. For occasional development, HC-110 and Ilford Rapid Fixer are a good place to start. And maybe add a drying agent (PhotoFlo). Newbies today are opting for the plastic Paterson tanks and reels because they're easy to load and pretty reliable. Easier methods? New55 has packaged a monobath which develops and fixes film in one bath.
Some of us are doing color at home too. See the sticky thread in the color forum for details.