You probably will not encounter any abstract math on your first go round, unless you finished the Calculus Series and Elementary LA/DEs. Some schools have a class that on proof writing that you take sometime before you begin the abstract stuff. If your school doesn't, you may want to check out:
How to Prove It: A Structured Approach by Velleman
The Elements of Advanced Mathematic by Krantz
These books are all right. But I personally don't think you get much out of "intro to proofs" books/courses without really having some mathematical theme. These books and courses that use these books dabble a little in set theory, a little in number theory, a little bit of geometry, a little of this, a little of that, etc...As far as I am concerned, the only plus to these courses is that you start getting used to notation, you learn the alphabet/vocabulary of proofs.
You might do better trying to tackle something like Dudley's Elementary Number Theory. At least you get more than just "a little" number theory, but still simple enough that you can cover the book and understand the material while getting used to doing math differently than the typical "Step 1, Step 2, Step 3 and Answer" as you are probably used to in High School math.
If you are taking the Calc series, and certainly when you take Elementary Linear Algebra, make sure you go out of your way to tackle the proof problems, even if you professor doesn't assign them.
Junior/Senior level math usually hits people like a brick wall (of course, this always depends on: the school, the professor, the book used, etc etc) because they are not used to abstract thinking (they've been taught to be great calculators). It understandable, not everyone taking Calc I-III and Elem LA/DEs are going to be math majors and the school usually only has a couple of different flavors of those series (like Calc for Life Science Majors and Calc for Physical Science majors/ Engineers, and even if the school has this, Physical Science and Engineers don't necessarily need to know how to do proofs).
In any case, long post still long...you won't get good at "proofs" until you do them a lot, and even then, proofs are usually easy only when you really understand the material, and sometimes not even then. Just practice, practice, practice
