If you could post up some examples, we could more easily pin point the problems and recommend solutions.
There where shadows behind the subject, and the subject wasn't lit up enough.
Was your camera turned to vertical/portrait orientation? If so, the shadows are probably a result of your flash being to the side of the lens, rather than directly above. This is where a flash bracket comes in handy.
If the subjects aren't lit up, it's likely that your i-TTL (auto flash metering) is picking up something as bright, which causes it to use less flash output. The solution is to adjust your FEC (flash exposure compensation).
I could crank up the noise, but what program do you guys use to take away noise?
Turning up the
ISO will give you more ambient exposure, which may or may not be a good thing, depending on the situation. It does give you more noise, but getting the right exposure really helps with that. There are plenty of programs to combat digital noise. Noise Ninja and Neat Image are/were a couple popular one. I just use the noise reduction in Lightrooom.
Another situation is outside, probably sunny, of a band ensemble. I'll probably have a tripod, and I'm guessing a f stop of 8-11.
An aperture of F8 to F11 should be good to help you get a deep enough DOF to get a group in focus, but you may need more, depending on the size/depth of the group and how close you are to them. Using a tripos is usually a good idea, but if your shutter speeds are fast enough, you probably don't
need it. The thing to watch for when shooting in the sun, is the dark shadows it causes on people's faces. You could try using your flash to brighten up those shadows (fill flash) but you may not have enough power in that flash to shoot at F11 outdoors in the sun.
Another situation is during a high school assembly, and there is a spotlight on someone. I haven't shot in this situation so what do you guys recommend for settings?
What you need to do, is find out what settings work for the light that is just in the spot light, not the surrounding darkness. So if you could get close and take a meter reading of someone in the spotlight...lock those settings in (manual mode). Then you can back up and shoot with those settings and have proper exposure. Keep in mind the shutter speed because it will cause blur if it's too slow. Crank up the ISO if you can't get fast enough shutter speeds.