Reddit reviews Combinatorics and Graph Theory (Undergraduate Texts in Mathematics)

Anything by Bollobas is great (graduate student level). Frank Haray's work is very accessible. Also, Combinatorics and Graph Theory is a brilliant introductory text for undergraduates or newcomers to the field.

Combinatorics and Graph Theory by Harris et al is a nice and short introduction to combinatorics, and even includes some interesting/exotic material not usually found in the introductory combinatorics books. Grimaldi is more comprehensive of course, but also more expensive.

I liked the Harris, Hirst, Mossinghoff text that I used for an undergrad class. It's a solid intro that builds from the very beginning, and at times has an almost irreverent writing style that references everything from Shakespeare to Kevin Bacon.