Why won't the stupid thing work?
You could be trying to hash out a tricky pivot on a hand-drawn animation or trying to adjust the rate of deceleration (also known as easing) on a tween, but it's not behaving the way you expect and you just can't figure out why. I tend to have a basic methodology for approaching problem-solving that helps me work things out and save a little time.
1. Walk away from the problem. Seriously. Whether you take ten minutes or a whole day, walk away from it and stop focusing obsessively. When you get tunnel vision, you're incapable of thinking clearly and objectively about the problem, and let your anger and frustration guide you in irrational attempts at troubleshooting. Walk away, clear your head, and come back calm and ready to approach the problem logically.
2. Break down the steps to isolate the issue. You need a clear understanding of what you're doing if you're going to find the source of the problem. Maybe on that character pivot, it's not really the entire pivot that's wrong; it's just the way the character's skirt swirls with the motion. With that tween, maybe it's easing out just fine, but it's doing it too quickly. You need to clearly define the problem before you can solve it. Often just by identifying the real problem (other than "it just looks wrong") you've given yourself the solution.
3. Ask yourself if what you're trying to do is actually possible. Maybe you're trying to use the easing controls to make a symbol speed up, slow down, and speed up again between only two keyframes, when you'd need more than that for the varying stages of motion. Maybe you're trying to animate motion that defies the laws of gravity or doesn't comply with the expected way the material of the swirling skirt should behave, so it looks unnatural. You may need to adjust your expectations and rework your concept to something that's functional and feasible.
4. Experiment with different approaches. The method you trusted obviously isn't working, so try something new. You've animated the swirl of the skirt incorrectly? Go watch videos of ballroom dancing. Or do something even more unorthodox: watch jellyfish moving beneath the sea, and see how the flow of their malleable flesh mimics the same flow you're trying to animate, and adapt that motion study to your needs. Still not working? Switch to pants. So your tween is moving too fast? Instead of tweaking the easing, instead increase the number of frames or even adjust the frame rate. There's more than one solution to any problem, and if one isn't working, then try another.
5. When all else fails, Google. You think I'm joking? I'm not. Odds are someone else has had the exact same problem, and either posted about it and gotten an answer, or described their own solutions. You don't know the number of times Google has saved my butt with a tricky ActionScripting problem, especially when adapting from 2.0 to 3.0 and often slamming my head on my desk repeatedly trying to get a class to work properly. Even if no one else has an answer to the problem, you'll likely find forums and other social areas where you're welcome to post questions and ask for help.
That's my basic methodology for approaching problems with my animations. It seems obvious, but you'd be surprised how many people have problems stepping back and working out a logical approach to troubleshooting. 99.9% of my problems can be solved this way, and often when I try alternative methods I end up with something better than I originally planned. For the other .1% of the problems, well...let's just say it's a good thing I don't keep my computer near any windows, especially since I live on the third floor.