I'm an english major with a vivid interest in mathematics,I've read and enjoyed What Is Mathematics? by Courant and Robbins (does this count as some background?),and I've decided to begin a serious study of mathematics, I've looked up many books and judged them by the reviews on ams ,maa and zbmath,and narrowed it down to the following:
Calculus/Analysis: Mathematical Analysis I,II By Zorich(what are the exact prerequisites for this one?) ,Problems in Mathematical Analysis by Boris Demidovich .
Complex analysis: Visual complex analysis by Needham.
Abstract/linear algebra: A Course in Algebra by E. B. Vinberg,Algebra by Michael Artin.
Geometry: Modern Geometric Structures And Fields by Novikov and taimanov.
Topology:O.Ya. Viro, O.A. Ivanov, Kharlamov and Netsvetaev, Elementary Topology: Textbook in Problems.
Differential Equations: the books by V. I. Arnold.
Methods in physics:Mathematical Methods of Classical Mechanics by V. I. Arnold ,A Course in Mathematics for Students of Physics by Bamberg and Sternberg.
Do you have any comments on the choice of books ? any better suggestions? any suggestions for other subjects ?
A good book for a first look at analysis is Bartle's real analysis. A very good introductory linear algebra text is Linear Algebra Done Right, by Axler. Another good linear algebra text is finite dimensional vector spaces by Halmos. For basic topology, nothing beats Munkres. Another good abstract algebra text is Basic Algebra 1 by Jacobson; another is Hungerford. You should also learn some number theory. For number theory, I'd say Davenport's Elementry number theory or Strayers.