Strategies for becoming unstuck in understanding TCS
Abstraction is pretty much bread and butter in computer science but unfortunately it is hard to teach explicitly.

In my opinion, understanding concepts is more important than being able to mechanically calculate or prove stuff. Sure, you need to know your way around some elementary methods, but the meat lies elsewhere.

First of all, you have to grasp the content to some extent. To this end, I have found it useful to ask the following question whenever something is unclear to you:


Why are we doing this?
What are we going to use this for?
What similar things does this relate to?
How do other sources explain it?
What exactly do I not understand?


After you have answered these questions (or discovered follow-up questions and treated them the same way) and still have problems, go to your teachers (or here). By now you should be able to formulate a focused, precisely formulated question; answering such questions is your teachers' job (and StackExchange's philosophy).

Other than that, it is exercise and experience. Try to reproduce proofs after having read them; take care to not learn them by heart but distill the important ideas from them. After some time, you should be able to reproduce all basic proofs by filling in gaps between the major steps. Even later, you will begin to see patterns in statements and proofs. This is how people look at a statement and say "Oh yeah, sure, use method X with theorem Y and then just use Z to get what you want.". It is pattern recognition fueled by years of training. Be patient.

As for basic exercises, go and find text books with some. Off the top of my head I can refer to Concrete Mathematics by Graham, Knuth, and Patashnik. This book is not only a precious toolbox for computer scientists, it also contains loads of exercises with solutions (!). Remember to attempt to solve them before looking up the answers and to reproduce answers you had to look up. 

Another useful book is Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein. Included is a sizeable chapter on mathematical basics. It also contains many exercises; solutions are available via the linked page (Supplemental Content). There is also a video lecture by one of the authors which may go nicely with the book.

For introductore lectures regarding proofs, have a look at Linear Algebra Proofs on Khan Academy. I have not watched them, but hopefully they are both basic and helpful. There are many more proofs on Khan Academy; I just figure that linear algebra proofs might fit computer science best. Do no hesitate to watch others, too.