First of all, why not go ahead and check out the math library of your university. Go there, sit on the floor and take books of the shelves and just skim through them.
Also, I noticed that you don't mind if there are problems in the books. That's good because you are wasting your time, to be honest, if you don't work through every problem on your own and treat doing the problems as more important than reading the sections.
Alright moving onto specific recomendations:
Conjecture and Proof, M. Lackzovich (this is a great into to doing proofs, has problems, and is wonderful for self study at an amatuer level)
A survey of modern algebra, birkhoff and mclane (I strongly recommend this book as an intro to modern algebra, it's very clear and easy to work through! Plus it will help you with proofs in a gentle way)
Differential Geometry, stoker (this is what i learned the subject from, I could not possibly recommend this book more for self study of this subject)
Principles of Analysis, rudin (this is a 100% no brainier, everyone else recommended it too)
Mechanics, L.D. Landau (course of theoretical physics vol 1)
Introduction to Analysis, whittaker (first edition, the second and third editions are not as good!)
Complex Variables, Fischer
Princeton Lectures on Analysis Vol 1 and 2, Shakarchi and Stein
Introduction to analytic number theory, apostle
Theory of Numbers, serpinski (this guy is from the older school, but its written really really well, and it is always nice to study the masters)
Lectures on Ramanujan, G H Hardy
Introduction to analytic number theory, apostle
Disquisitions Arithmatic, Gauss (just try to get as far as you can!)
Abstract Set Theory, Fraenkel (this author has the axioms of set theory named after him, and this book is a really fast and fun read)
that should be enough to keep you busy for awhile =)