The nice thing about papercraft is that it really isn't a very expensive hobby. No expensive model kits to buy, no exotic materials needed. Many of the things you'll need you might already have at home. While not all of the materials below are necessary, if you get into papercraft
Hobby Knife - The most important tool in your papercraft arsenal. Get a precise knife with a fine blade.
Scoring Tool - Not always necessary, but makes scoring and folding paper 100 times easier.
Glue - White glue works fine. I prefer wood glue myself.
Paper - You'll need slightly thicker paper than what you might normally use for printing documents.
Cutting Mat - Not necessary, but they make your life much easier. No cutting into your precious wood table or shredding up stacks of newspapers.
Inkjet Printer - You can use any kind of color printer, but inkjets seem to work the best.
I'll first go over each of these pieces of equipment in more detail, and then get to finding papercraft models, printing them, and technique for actually cutting, folding and gluing the parts together.