Good textbooks on combinatorics for self-study
One book I'd highly recommend is Peter J. Cameron's
Combinatorics: Topics, Techniques, Algorithms
The first link above is to site for the book, which includes multiple resources, including links, solutions to problems (good for self-study, etc.), additional exercises and projects. I used it in an early graduate "Special Topics" class on Combinatorics. (It's geared to be "basic" for grads, advanced for undergrads. Indeed, the text is designed in such a way as to provide ample material for a two-semester sequence, for self-study, and as a reference for future needs.) I have since self-studied the text on my own, to cover material which simply could not be covered in a single semester.
The text is loaded, and expansive in its coverage: dealing not only with combinatorial "content", but also combinatorial techniques, proofs, as well as algorithms for those with computational interest in combinatorics etc...and Cameron himself suggests which chapters to study for different aims.
You can peruse/preview the book at Amazon: Take a look inside, and also through the top link.
Table of Contents
Preface
1. What is combinatorics?
2. On numbers and counting
3. Subsets, partitions, permutations
4. Recurrence relations and generating functions
5. The principle of inclusion and exclusion
6. Latin squares and SDRs
7. Extremal set theory
8. Steiner triple theory
9. Finite geometry
10. Ramsey's theorem
11. Graphs
12. Posets, lattices and matroids
13. More on partitions and permutations
14. Automorphism groups and permutation groups
15. Enumeration under group action
16. Designs
17. Error-correcting codes
18. Graph colourings
19. The infinite
20. Where to from here?
Answers to selected exercises
Bibliography
Index.
Also, for a sample of Cameron's writing style, here's a pdf format of his Combinatorics Notes (also available at Cameron's home page, which is accessible from the book's site to which I've included a link at the top this post).
One other text that might suit your needs is Introduction to Combinatorial analysis by John Riordan, considered by many to be a "classic*.
A Path to Combinatorics for Undergraduates: Counting Strategies [Paperback] Titu Andreescu (Author), Zuming Feng (Author)
Authors take a problem and start solving it. You start following solution and then all of a sudden they say "Not so fast my friend.. not so fast..." pointing out how that logic was faulty and then give the actual solution. This gives me kicks.
Richard Stanley has made his Enumerative Combinatorics, volume 1, available free for personal use (for awhile) on his website at http://math.mit.edu/~rstan/ec/ec1/. There's a new edition coming out and he's looking for possible corrections.