I write my dreams down in my regular journal. (I once tried to keep a separate dream journal, but I do better keeping everything in one place.)

During the writing I usually start remembering additional details, even if I remember only snatches at first. I'll go through periods where I remember nothing at all, followed by a gaggle of remembered dreams. My journal is unlined, so sometimes I'll also draw scenes from a dream -- the placement of people and objects. My drawings are mediocre at best, but they're meant to be recorded memories, not art.

Good luck and have fun!