To begin, you do have to understand the definitions. It's impossible to prove anything if you don't really understand the question.
Linear algebra has its own terminology, and many a student has gotten so tangled in it they could not proceed well. Besides the statement of the definitions, you have to understand what they mean and how they relate to other definitions. The best way to accomplish that is to work a lot of basic problems, dealing with vectors, matrices, systems of equations.
Next, there are a few fundamental theorems which come up over and over. There is a laundry list of about 12 statements which are all equivalent to saying that a set of vectors is independent. You should get your hands on such a list, and then make sure you know why each one is equivalent. If you can't find such a list (which I provided my own students), why not ask here. I'll bet you wind up with 30 equivalent statements. (Not that you necessarily want all 30, I know).
The next step would be to work many, many problems about vector independence and basis. If you get stuck, get help, and make sure you understand the answers.
The next big linear algebra topic is "spectral theory" which is about eigenvalues and eigenvectors. Study that separately, and again work a lot of problems. Stay away from generalized eigenvectors until you have mastered the basic material.
This short list is only a start on the subject, but it can help you understand what is most fundamental. I think it is always best to solidify foundations, after which moving forward is much easier.
Regarding books, you now have several recommendations. I find that different approaches work better for different people, so that someone may love a book that makes no sense to you at all. Try out various chapters that have been published online and see whose style appeals to you. A book that provides many problems of varying levels of difficulty would be the best.
The best way to learn the subject is to work many problems. Along the way, you'll get a sense of how to approach linear algebra proofs.