A good first project, which will be simple, yet very informative, would be to construct a uC based "chaser light" system (similar to christmas tree chaser lights, but with LED's). Add a push button which triggers an interrupt to speed up the chasing, and a push button on an I/O line to slow it down.
By the end of it, you'll have a decent grasp of uC basics, and have learned an important lesson about interrupts and polling.