It is a good thing to try different books, in my experience as a self-learner I found that a lot of traditionally aclaimed books are incredibly hard, there's always an author that can help you to grasp core ideas easily, for example, in calculus I read a little of the calculus made easy by Silvanus Thompson.
Springer has a lot of titles on proofs, and there are also some books you should look:
Bridge to Abstract Mathematics: Mathematical Proof and Structures - Ronald P. Morash
- This is a really nice book, it made a lot of things on set theory, logic and proofs a little easier to me.
How to Solve it - George Pólya
- This is a classic book, I guess you must be aquainted with it.
HOW TO PROVE IT: A Structured Approach - Daniel J.Velleman
- I'm about to read this one, it seems to have a nice purpose.
Linear Algebra As an Introduction to Abstract Mathematics - Isaiah Lankham, Bruno Nachtergaele & Anne Schilling
- I dont remember how I found this book but perhaps it may be of help to your case,I found it in my library and it seems to be a mix of
Linear Algebra and proofs, it seems nice for your case.
There's a class of books that may be also helpful for your case, the transitions to advanced mathematics:
Mathematical Proofs: A Transition to Advanced Mathematics - Gary
Chartrand & Albert D. Polimeni & Ping Zhang
A Transition to Advanced Mathematics: A Survey Course - William
Johnston & Alex M. McAllister
A Transition to Advanced Mathematics - Douglas Smith & Maurice Eggen &
Richard St. Andre
Also some references on real analysis:
A First Course in Mathematical Analysis - David Alexander Brannan
- I really loved this book, as the author says in the preface: Changes in the school curriculum over the last few decades have resulted in many students finding Analysis very difficult. The author
believes that Analysis nowadays has an unjustified reputation for
being hard, caused by the traditional university approach of providing
students with a highly polished exposition in lectures and associated
textbooks that make it impossible for the average learner to grasp the
Introduction to Real Analysis - Robert G. Bartle & Donald R. Sherbert
- This is also a good one, a little harder than the first one but still nice.
I was thinking about this answer and I reminded of one thing that I took a lot of time to understand: the concept of the best book. The best book is the one that makes you learn. In an analysis course, most people will tell you to read Rudin's book, for calculus they'll say you to read Apostol's book, this is kinda invalid and it really depends on your background in mathematics, it's good to remember that such books were written in different circumstances and that the authors presume that the readers know some things. I'm not discrediting these books, they're nice, but it will be much better if you learn with something easier and then try to read these hard books later. Always try to find books that are compatible with your mind, this will make your mathematical experience a lot better. You can also try to read topics spoken by different people: Having trouble with one author's definition of sequence? Try to read the definition by other author, I'm doing this with the books I mentioned: When something is hard on Sherbert's book I read what Brannan has to say about it.
I hope it helps.