What exactly are these like? I've had symbolic logic and we did proofs in that class. Are math proofs anything like these? We had some premisses and a conclusion and had to use valid rules to get the conclusions.

It really depends on how you want to prove things. When we first learned proofs we used all the symbols like "for all" and "there exists." As we got more and more into it, proofs began to use more words instead of symbols. I still use symbols just because I am lazy.

You have to remember that the point of a proof is to show, without a doubt, that the conclusion follows irrefutably from the premises and axioms. A proof for one theorem you do in an introduction to proof writing is different from a proof for the same theorem you might give in a senior level course, too. This is because at some level, it is assumed that the proof writer and the proof readers both understand the same basic understandings and so it's a waste of time and energy to restate the same elementary information.

I'm not sure how difficult the proofs you were doing in your logic classes were. In math, you still start with assumptions (i.e. axioms) and prove things with those assumptions, but the arguments are less formal in terms of usage of symbols, so more plain English arguments are included. The emphasis is more on the logic itself and less on the notation of logic, aside from mathematical notation of course.

Since I haven't studied advanced logic as a philosophy, all I can say is that there are many provable things in math where there's simply no obvious way to go about completing the proof. Quite often, you have to recognize patterns and make lucky guesses at the method first, then prove that your lucky guess solves the problem. So if you prove something in math that hasn't been proven before, this is recognized as a very difficult achievement and you'll at least have your name recorded in history next to your discovery.