The bulk of the work in this course will consist of assignments that require you to write proofs. Assignments of this type are different from those you have experienced in lower level courses like calculus, and for most people a somewhat different approach is needed.
First, you should understand that writing a proof is in some ways similar to writing a short composition in an English class. Most likely, you will produce one or more drafts of your proof, each more polished than the preceding ones, before you have a finished product suitable to hand in. The number of drafts will vary with the complexity of the proof, but you'll probably have at least one draft version.
Second, just as writers experience "writer's block", you should not expect to have a proof occur to you in one sitting. Most of the time, there will be a period when you are thinking about a proof, but do not yet have a complete idea how to do it.
In problem-oriented courses like calculus, you can usually complete a problem within, say, an hour of starting to work on it. You should not expect this when you are writing proofs, because proofs require some "incubation" time where you are formulating the proof before you actually begin writing it. This is a creative process, and as such the time required can vary quite a bit. Sometimes a sketch of the proof will occur to you quickly, and other times it will take several days or even longer.
A good approach is to consider each problem to be an interesting and challenging puzzle to be solved. Solving a puzzle can be a lot of work, but in spite of this many people enjoy doing them, and they find solving a difficult puzzle to be a source of satisfaction. People who choose to study Mathematics often fall into this category. Writing a proof is very much like solving a puzzle, and completing a challenging proof can be a very satisfying experience.
An undergraduate Mathematics professor of mine said that the best reason to study Mathematics is because it's "a fun game". Most of the professional mathematicians I have known seem to feel this way, and if you can cultivate this attitude it will make things easier.
Because of the incubation time phenomenon, it is important that you read the problems you are assigned and begin thinking about them as soon as possible. A very good approach is to commit the problem to memory and think about it off and on during the day. It is a remarkable fact that from the time you first encounter a problem, your brain may be working on it even if you are not aware that it is.
If you are stuck on a proof, it's a good idea to put it aside and do something else. Very often, when you come back to it, you will have some new ideas on how to do it. A technique that works for many people is to review the problems you are working on before you go to sleep. It is very common to wake up in the morning and find that a solution occurs to you. Conversely, procrastination is a very bad idea. If you wait until just before an assignment is due, there may not be enough time to come up with the proofs, and the anxiety over completing the assignment on time can interfere with the creative process.
You may work together on these proofs, and in fact I encourage you to do so. You can share ideas on how to do a proof from your classmates, but of course the final product should be your work. Because of variations in wording and organization, it is highly unlikely that two people will independently come up with identical proofs, even starting with the same idea of how to do the proof.
When you begin working on a proof, a good way to start is to read the statement you are trying to prove, and ask yourself if you think it is true. If you think it is true, ask yourself why you think it is true, and see if you can orgainze these thoughts into the beginnings of a formal proof.