Hi! This is a very useful question. Getting the right books is very important. But Math books are expensive, so I suggest you to buy international edition (you can look on abebooks.com) or check your library. Below is a list of books which will definitely prepare you for a rigorous graduate program in Mathematics. It's divided into subfields of Math and within each subfield, it's sorted from easy to hard (from beginning university level to finishing university).
Analysis:
- Principles of Mathematical Analysis (Rudin)
- Calculus on Manifolds (Spivak)
- Fourier Analysis, Complex Analysis, and Real Analysis (3 book sequence by Stein and Shakarchi)
- For Complex and Real Analysis, you could use Rudin's Complex and Real Analysis as well.
- Functional Analysis (Rudin)
Differential Geometry:
- Do Carmo's Differential Geometry and Riemmanian Geometry (Do Carmo)
- Introduction to Topological Manifolds (John M. Lee)
- Introduction to Smooth Manifolds (John M. Lee)
- Riemannian Manifolds: An Introduction to Curvature (John M. Lee)
Topology:
- Topology (Munkres) -- very good introduction.
- From Calculus to Cohomology (Madsen and Tornehave) -- a good introduction to Algebraic Topology through differential forms; can be read after Calculus on Manifolds by Spivak.
- Algebraic Topology (Hatcher) -- very geometric, but you need to know some algebra first (see the algebra list below).
Algebra:
- Abstract algebra (Dummit and Foote) -- Easy to digest, but I much prefer the following
- Algebra (Lang, GTM) -- very amazing introduction, a bit terse.
- Commutative Algebra (Atiyah and Macdonald).
- Basic Algebraic Geometry (Shafarevich), or Algebraic Curves (Fulton, free online) -- good intro to algebraic geometry.
- Algebraic Geometry (Hartshorne) -- difficult book, but worth the effort if you want to study algebraic geometry.
Number Theory:
- Algebraic Number Theory (Neukirch) -- good book, but you need to at least read the first 4 chapters of Lang's algebra before starting. The section on Analytic Number theory also requires a good understanding of Complex Analysis.
Of course, you don't need to read everything up there. What you need to learn depends on what math you want to do in the future. I think it's good to see a professor and talk to him in person to see what you really want to do.
I hope that helps!
Edit: added suggestions by Andrew L.