I've been learning a lot about shooting various sports and concerts over time, and wanted to share some of what I've learned/others have told me about here. I'm not going to go over the whole "be respectful and considerate thing" since I've mentioned that before, and honestly, it should be obvious. Most of this stuff will apply across the board to both.
I need to give kudos to Alan Hess here, some of this stuff he mentions in his Photoshop World Concert Precon, and in his Kelby Training class, but I don't remember which ones. It's one thing to be told, and another thing to see it in reality.
1) Try to get away from other photographers
This is more true in concerts, because in a lot of sports, you have a set place to shoot from. Like for the Colorado Rockies, you can shoot from the 1st base photo pit, the 3rd base photo pit, or the concourse, nothing in the stands. But for high profiled concerts where there might be a small photo pit, and a lot of shooters, try to not get caught up in the fray. When I shot Mayhem Festival, everybody started on the right side for Megadeth, and was packed like sardines there, and I started on the left, and had a lot more breathing room.
2) Watch your background
In a lot of cases, you really don't have much of a choice. If you're shooting a high school soccer game on a field which is surrounded by a fence with houses, the best you can do is make sure the background is as out of focus as possible. However in a lot of cases, just moving around and switching angles is good.
I really like this snowboard photo, but the lift is in the background. Something I should've been more aware of when shooting it.
3) Shooting wide vs shooting tight
I know people who shoot very tight on their subjects so they don't have to crop. I also know people who shoot wider on their subjects and crop just a little bit if they have to. Neither is right or wrong, just be aware if you shoot tight, to be careful of things you might cut off, like the guitar stock. Not that I'm saying you never want to cut off the guitar stock, but there's ways to make it look good. Personally I like shooting just a tad wide since I do have a bad habit of cutting off the top of the stock when I'm trying not to.
4) Don't crop at a joint
Cropping a person tighter is fine, just don't crop directly on top of a joint (knee, shoulder, elbow).
5) Microphone stands and microphones suck
That's it. They're a necessity of shooting at a concert, but they really suck because they're always in the way of something. Just be careful how you shoot the performer that's behind the stand, and you should be ok.
Here's an example of a really horrible angle to shoot the singer at since you can't see his face.
6) Watch your shadows
More of a function of concerts than of sports, but the lighting can play some weird tricks on the performer's face. A lot of times if the singer is right up against the mic, you'll get a weird shadow on their face. If you wait until they hit a high note and pull back just a little bit, you'll get less of a shadow.
While I'm not a big fan of this photo regardless, it shows what the shadow can do. In this case, it's cutting part of his face in half.