I'm going to repost my e-mail response to you here, in case others are interested:
As a first step, I've bought Discrete Mathematics book and reading it for last two months.
Good.
What should I move on to once I understand Discrete Maths? Straight to algorithms? How do I approach algorithms?
I'd actually start with data structures, but this can be tricky as I don't know of a really good standard datastructure textbook. We learned mostly from professor's handouts. We did have a textbook - Data Structures and Algorithms in Java by Goodrich and Tamassia (http://ww0.java4.datastructures.net/) - but its quality was spotty.
I also learned some from "C++ Components and Algorithms" by Scott Robert Ladd (http://www.amazon.com/Components-Algorithms-Scott-Robert-Lad...). This is old - it's what I first picked up when I was 15 and playing with MUDs - and a little basic, but it got me started.
Cormen also has a lot on data structures, but the level of presentation is pretty high. It may be easier to combine it with another source.
I mean should I understand every proof written in Cormen's book? I'm confused as to how should I approach this.
This will take you a long time - I don't understand every proof in Cormen's book, I just go back to them as necessary. I'd start by understanding what's going on under-the-hood when you use standard library datastructures - linked lists, vectors, heaps, trees, hashtables. Once you've got that, go on to understand the variations of these, eg. the half a dozen or so different collision-resolution strategies used in hashing.
When you've got that, take a look at some of the more advanced data structures in there, like binomial and fibonacci heaps, skip-lists, B-trees, etc. And somehow make time for graph algorithms, they come in handy.
What other math books should I be reading?
You should make time for Structure and Interpretation of Computer Programs (http://mitpress.mit.edu/sicp/). Other good theory books: Concepts, Techniques, and Models of Computer Programming by Peter van Roy, and Types and Programming Languages by Benjamin Pierce. As for math - it's good to know some set theory and mathematical logic, but I wasn't a math major so I don't have good recs in that area.
Being a mathematics major as well as CS, can offer a recommendation of a book for each of set theory and mathematical logic. Each were the basis for the course I took on the topic, and recieved pretty good reviews on Amazon:
Karel Hrbacek and Thomas Jech, Introduction to Set Theory. ISBN-13: 978-0824779153.
Herbert Enderton, A Mathematical Introduction to Logic. ISBN-13: 978-0122384523.
The Enderton book was what we used for my Mathematical Logic course, and it was pretty good. Thin, but a pretty clear and (apparently) comprehensive presentation.
I also learned some from "C++ Components and Algorithms" by Scott Robert Ladd (http://www.amazon.com/Components-Algorithms-Scott-Robert-Lad...). This is old - it's what I first picked up when I was 15 and playing with MUDs - and a little basic, but it got me started.
Cormen also has a lot on data structures, but the level of presentation is pretty high. It may be easier to combine it with another source.
Also, the C2 Wiki has some great pages on data structures and algorithms: http://c2.com/cgi/wiki?DataStructures
This will take you a long time - I don't understand every proof in Cormen's book, I just go back to them as necessary. I'd start by understanding what's going on under-the-hood when you use standard library datastructures - linked lists, vectors, heaps, trees, hashtables. Once you've got that, go on to understand the variations of these, eg. the half a dozen or so different collision-resolution strategies used in hashing.When you've got that, take a look at some of the more advanced data structures in there, like binomial and fibonacci heaps, skip-lists, B-trees, etc. And somehow make time for graph algorithms, they come in handy.
You should make time for Structure and Interpretation of Computer Programs (http://mitpress.mit.edu/sicp/). Other good theory books: Concepts, Techniques, and Models of Computer Programming by Peter van Roy, and Types and Programming Languages by Benjamin Pierce. As for math - it's good to know some set theory and mathematical logic, but I wasn't a math major so I don't have good recs in that area.