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
The Mathematical Association of America (MAA) has got a rich collection of classic books under Doclani Mathematical Expositions. I would suggest you following:

$1$. Mathematical Gems Series ($3$ Volumes) By Ross Honsburger.

$2$. Linear Algebra problem book By Paul R Halmos.

$3$. Euler: Master of us all By William Dunham.

Some other texts:

$1$ Introduction to Commutative Algebra by Atiyah and MacDonald.

$2$ A book of abstract algebra by Pinter.