Graduates

For most people who take it, Discrete Math is completely different from other classes you've taken. The point is not to know the answer to a question, but to learn the tools you need to prove mathematical facts. The best advice is probably to not hurry through problems, both on homework and the tests. If you do work quickly, make sure you check for argument for holes afterward (i.e. did you make sure the denominator's not 0? that kind of thing).
The second piece of advice I have is to always use what you know. When you just see the statement: "Prove ..." it's easy to psych yourself out until you have no idea where to start, but just try a few of the things you learned in class and you'll probably run right into the correct answer.

Don't be afraid to ask questions. The professors are there to teach, but they can't help you if they don't know where you're confused. A quick question in class, a short conversation after class, may be all the help you need to turn that complicated problem into something you can do easily.
Think outside of the box. Math is it's own language, and occasionally it does things that don't seem to make sense. However, if you dig into it enough, you can always find out why it works.
The best way to make sure you really understand something is to teach it, so use your friends as practice dummies. Bounce the concepts you learned in class off of them. If you can get your friends to understand it well, chances are you know it pretty well yourself.
Arrange with some classmates to study together; one of them probably understands something you don't and can help you figure it out.
All in all, just stick with it. Don't be discouraged if you don't get something the first day it comes up, but don't sit there idly waiting for it to make sense either. Use your resources, ask questions, read the book, and even use the internet to find alternate ways of describing the concepts you're learning in class. Best of luck to all of you!