My favourite models to make are the Frog and Modular origami. Modular origami involves making a number of separate pieces which are then assembled into a larger model, for example six or twelve pieces making a cube. The sonobe system is a good model to start with.
Modular cube (twelve pieces)

image from ceciliacotton.ca
Paper
Square paper is easy to make. Use a sharp craft knife and a metal ruler. Get some A4 copy paper, or a few pages of A4 junk mail. Taking one sheet of paper, fold the top-left corner down to meet the opposite edge, about a quarter of the way up from the bottom right-hand corner. Use the ruler and knife to cut the resulting strip off the bottom. If you don't have a knife handy, neatly fold the strip back and forth a few times, and then carefully tear. Unfold the triangle and there's your square.