How about this: Start "simple", then move from there. I'll actually give you a problem to solve; while it sounds simple, it is anything but:
Build a robot which can navigate from room to room in a house, avoiding obstacles in its way. How it avoids these obstacles is up to you (ie, what sensors you want to use), but the robot needs to be able to:
a) know where it is in the house
b) know where it was
c) know when the immediate environment changes, and adapt
Room beacons or other external novelties are allowed; You can have a "built in" map, or it can build the map from experience (i'd suggest the latter). You should be able to have the robot randomly move from room to room, and also be able to "tell" the robot a "trip" (go to room A, then B, then C, then return - or something of that nature), and it should be able to follow that plan.
As I noted before - this sounds easy, but it is far, far from it. In fact, it isn't a completely solved problem. The problem (which doesn't have a general solution yet - but there is much study on it!) is called SLAM (Simultaneous Localization and Mapping):
You could easily spend a few years of learning and research on this one area alone; it has many, many practical uses in robotics (and a few other fields; but mainly robotics).