New Mathematical Library books #5, 11, 12, 17, 25, 27, 29, 31, 33, 38, 40, 42
Work LOTS of problems from old tests!
Neither calculus nor linear algebra are needed, but if you're ready for these subjects then by all means continue your studies in math by studying these subjects (when you have the appropriate background), as life does not end when you're finished with high school math contests.
(added next day) In the time since I wrote the above (which was done very quickly, but it seems fine on rereading now), I've looked through my books at home and found 4 that merit mention.
1. Posamentier/Salkind, Challenging Problems in Algebra
This is a cheap Dover Publications (1996) reprint of a book originally published in two thin volumes in 1970 and reprinted as one volume in 1988. The difficulty of the problems (all of which have solutions) is slightly higher than the typical (strong or honors level) U.S. Algebra 2 course, which makes it an excellent place to start if your background isn't very strong and you're looking for easier AHSME level problems that stay within approximately U.S. high school Algebra 1 and 2 material.
2. George Polya, How to Solve It
This is a universally recommended book for contest math that was first published in 1945. The 2nd edition was published in 1957, and the 2nd edition was later reprinted in 1988 and 2004. In my opinion, the book is better for the cognitive strategies it lays out than for specific mathematical "tricks". However, because of this the value of the book will increase as you learn more mathematics and get more experience in contest problem solving.
3. Arthur Engel, Problem-Solving Strategies
This is an encyclopedic book (first published in 1998) that contains a huge number (over 1300) problems, all of which are either fully worked or provided with a substantial hint and/or solution outline. This is the most complete and extensive book on contest math problems that I am aware of.
4. Steve Olson, Count Down
This book was first published in 2004 and it gives a "popular" account of the 2001 U.S. IMO team and that team's IMO performance in 2001. Included are discussions, for each of the 6 IMO problems from 2001, on how various contestants solved the problems.