The following texts were very successful in helping me make the transition from high school mathematics to university level mathematics. When I left high school in the UK system (A-levels) I had a reasonably thorough grasp of calculus, trigonometry, geometry, school level algebra, statistics and mechanics (these were covered in the A-level maths and further maths syllabi, if you're familiar with these at all).

General reasoning:

1. Copi and Cohen, 'Introduction to Logic'. A basic introduction to logical reasoning. Despite having studied as much mathematics as I could at high school, I was never taught to understand the logical structure of proofs. Reading this book helped me to begin reading proofs in undergraduate/graduate mathematics books.

Analysis:

1. Bartle and Sherbert, 'Introduction to Real Analysis'. This is useful for picking up the basics of real analysis at an elementary level and is also useful in learning to read proofs of elementary results. I found that this phase of my education was a bit tedious and that I was wanting to get ahead to a more advanced text. It is a useful complement to item 3 below.

2. Kolmogorov and Fomin, 'Introductory Real Analysis'. This is an excellent book for learning analysis and was used in some honors level analysis courses for students who were familiar with reading and writing proofs. I would emphasize that reading Copi and Cohen first is an imperative.

3. Walter Rudin, 'Principles of Mathematical Analysis'. Harder to read than Kolmogorov and Fomin, but it contains beautifully written proofs and is a great text to model your own answers on.

4. H.L. Royden, 'Real Analysis'. An excellent book to learn from on measure theory, integration and basic functional analysis (classical L^p spaces, etc).

5. T. Gamelin, 'Complex Analysis'. A good book on complex analysis with lots of motivating examples.

Algebra:

1. Allan Clark, 'Elements of Abstract Algebra'. This is a Dover publication and is rather cheap (probably around \$15 or so now). It consists of a hundred or more articles; you are given definitions and the proofs of a few important theorems. Everything else is an exercise. I believe this book did the most for me in helping to build intuition for abstract algebra.

2. Hoffman and Kunze, 'Linear Algebra'. A classic book on linear algebra which I do not think has been surpassed. It provides thorough and well written proofs. I believe this is a preferable text to Axler's book, unless you are somewhat heavily inclined towards analysis and/or do not enjoy doing computations.

3. Serge Lang, 'Algebra'. This was my second book in abstract algebra, after Clark's. It is beatifully written, which is why I prefer it to Hungerford's book. As another answerer mentioned, the typesetting in Hungerford's book is also somewhat off-putting, and Lang's book does not suffer from this defect.

Topology:

1. Munkres, 'Topology'. A clearly written text with a good supply of examples.

Lastly, I would second the idea of purchasing access to a university library with a good collection of mathematics textbooks.

Best of luck.