I find that studying for analysis is unlike other math classes that I've taken. I dedicate a lot of time to studying for it, but it seems like no matter how much time I put into it I am not getting better. I think I'm studying in the wrong way, but I'm not sure what the right way is. It seems as if every new problem that I try, I have no idea how to go about it and I need to look at the answer. Also when reading the textbook (Rudin) I get stuck on something in a proof and don't know what to do from there to understand it. I know the proofs of the theorems are important to understand but I get stuck very often. Any advice?
Unlike algebra or topology, doing analysis is not merely a matter of deducing proofs directly from the definitions, although that's clearly a big part of it. It's much more like it's source calculus, in that direct computation is a critical part of solving problems. Unlike calculus, though, the kinds of "computations" you do in analysis aren't simply rearranging equations or substituting trigonometric identities. In analysis, you have to understand much more clearly what's going on in order to do the computations, you can't do them blindly.
For example, the main tool of analysis are basic inequalities like the Cauchy-Schwarz inequality and in a limit proof, you need to know which one to use to produce the right choice of epsilon for every delta. The huge mistake a lot of mathematics programs today make is that they teach students in basic calculus that these 2 types of problems are completely different-that substituting a $u$ in an integral is a completely different and reasonable problem for humans to do and calculating a limit using epsilons and deltas is an alien problem for geniuses. It's crap-the only difference is that in the latter, you can't just mechanically solve the problem, you have to think about it. To me, this is one of the hidden reasons students struggle with analysis even though they did great in baby calculus-they're conditioned to think it's something only geniuses can do. It is more difficult than the calculus they're used to, but it's not a different procedure-it's the same in kind. It just needs to be understood better before proceeding.
I think your main problem is you need more practice with inequalities and epsilon-delta arguments. A very good source to practice with is R.Burns' Numbers And Functions, which gives many good problems with hints. You also need to strengthen your knowledge of inequalities and how to work with them. A very good and cheap source for this is Karzinoff's Analytic Inequalities, now available from Dover. I think once you strengthen this background, a lot of your problems with analysis will begin to go away.