As others have indicated, it would be much easier to answer this question if you gave a better indication of the level at which you are currently studying mathematics/trying to prove theorems. (High school, undergrad, doctoral student, professional, ... ?) Nevertheless, here is some general advice, which is similar
to the advice I give my own students at the advanced undergrads/beginning doctoral student level.
Let's begin with what professional mathematicians do: they prove new theorems, results that have never been proved before. Doing this has one obvious difficulty: until the theorem is proved, you can't even be sure that it is true.
So you have the problem of trying to guess at what might be true before you can
be sure that your guess is correct. The way that people do this is by some kind of intuition built up with experience. As a general rule, it is probably best to postpone trying to do this until you have had time to build up the necessary intuition and experience; typically that time would be some point during the course of your doctoral studies.
Prior to trying to prove new theorems, the best thing to do is to practice proving theorems which you already know are true. One source of these is
homework problems assigned in the courses you are studying, but it sounds from your question as if you might be having trouble with those.
A good way to improve your theorem proving skills, then, is to do the following:
go back to a (theoretical) course you studied some time ago, and whose results you feel comfortable with. For a lot of people this would be a first course in
group theory, or maybe a course in elementary number theory. Then try to prove the major theorems from the course without looking at your notes or textbook.
The point of this is that you are hopefully fairly familiar with the statements
of the results, having used them a lot of times since then, and are probably reasonably familiar with the techniques of the course too. But on the other hand, you probably don't remember all the proofs exactly. So it is a good place to practice: your past experience with the course, and the advantage of having seen the results proved carefully before, should help serve as an intuition which can guide you in trying to reprove the results yourself.
If you can't work out how to prove a result after some time, you can look back in your notes or in the text. But try not to look at the whole argument. Just look to find the point in the argument where the author deals with the point on which you are stuck. Once you have seen what general technique they use to get past this sticking-point, stop reading! See if you can use that technique yourself to finish the proof. In this way, you will begin to learn the different techniques and how they are used in arguments.
This is the process I used (and continue to use) myself as practice for proving
theorems, and I highly recommend it. The more theorems you prove yourself, the better you will get at it! Once you have gone back over a course in this way, you will also understand the results and techniques of that course much better than you did before.