Question: In his book "Men of Mathematics", Eric Temple Bell repeatedly makes the point that a student of mathematics must read the classics.

My question is what are some classic books in mathematics ( Dictionary definition : judged over a period of time to be of the highest quality and outstanding of its kind.) that can be used by a high school/undergraduate student to start the study of higher mathematics? 

Some subjects I would like reference in particular, otherwise state any book you consider a classic, are:

1) Analysis
 2) Abstract algebra
 3) Linear Algebra
 4) Number theory
5) Combinatorics and Graph theory, etc.

Answer: 
1. Feller's first volume for Probability Theory.
2. Arnold's ODE for differential equations.
3. Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables for Complex Analysis
It's a bit more advanced than the topics you asked about, but Milnor's *Morse Theory* and Milnor and Stasheff's *Characteristic Classes* are astoundingly good. (There's a pattern here: Milnor's *Lectures on the h-Cobordism Theorem* is pretty good too!)

At a somewhat lower level, I find Spivak's *Calculus* (which many might argue is an introductory analysis book) pretty darned wonderful.