If you think you're going to want to experiment and get into photography, then definitely go for an SLR, with changeable lenses. These are too bulky to quickly grab and take surprise shots with though. If so, I'd recommend the Canon 350D
If you are content to play with some manual controls but be limited by the features of a smaller camera (regardless of what technology they fit into it, they're constrained by certain laws of physics), then go for an advanced compact camera. If so, I'd recommend the Casio EX-Z850 or Canon SD850.
Another type of camera that's becoming popular is for those that want more of the SLR features, with larger zooms and more controls, without the hassle that a SLR brings (changing lenses, etc.). These mega-zoom cameras, such as many in the Panasonic range, are very good too.
You're best to try them out first, and see which one you'd be happy with, in terms of size, and ability. A compact will never give you the range of playfulness that an SLR can bring, but it may well be enough for you, especially if you're just starting out.