Think about having a couple of really strong images of each event. Get the winners on the podium.
Winners on the podium--often easier said than done. Many (if not most) of the events I've done just kinda hand out awards and call it good without really setting up a shot. Step up and take charge and get the people to pause and pose for a photo. Move them around if need be (all step up on the top podium for a nice close up). If you just wait for it to happen....well, it won't. ;-)
A few sport specific thoughts-- in cycling a good slow shutter panning shot is often one of the most expressive, but you'll get a lot more misses than hits. If it's just "get a shot", crank 'er up and freeze the action. But if you have multiple chances and can afford the misses, go for a panning shot.
Don't crop runners at the knees or ankles--runners love to be able to see their feet, especially if you can catch them airborne.
Photographing swimming sucks--good luck. If you have limited time, try to catch the butterfly event(s) as those are some of the most dramatic shots.
You mentioned wanting to outdo the cell phone pics--one of the easiest way (IMHO) to do that is simply get the shots that they can't, which means more challenging situations such as freezing motion in a poorly lit venue.
I'll post a few samples below (not necessarily my best, just what I could dig up while here at the office)...