The best introductory book for "real mathematics" (theorem-proof style) that I've seen is Calculus by Michael Spivak. It is a large book, lucidly explained in great detail. It teaches insight and intuition, and has a very "chatty" style, as one of my professors once put it. Stay away from his other books, though. They are very advanced and leave much to the reader to prove.
That being said, I think you ask the wrong question. Don't just give a reading list. As a teacher, you should be doing the reading and teaching things to your students. Most people will not take well to a giant (or even small) list of math books to just read.
Basic group theory is very nice and has many accessible results. The book I used is by Fraleigh and is called "A First Course in Abstract Algebra." The first half of the book is about groups.
If you are interested in computer applications, "Simulation" by Stephen Ross is quite good. It is reasonably basic and certainly requires little calculus. Most of the assignments involve programs that can be written in 20 lines of python--probably more for C/C++/Java. It shows a nice example of how computers can be used for nontrivial mathematical applications (i.e., more than just adding numbers and computing derivatives/integrals that are "hard").
Other topics of interest are Probability (the dice kind, not the measure theory kind), Combinatorics, and basic number theory. I always thought Linear Algebra was pretty cool--as long as you don't focus too much on the boring mechanical junk like Gaussian elimination and stick more to the abstract notions of vector spaces, bases, eigenvalues, and spectral theory. If you are feeling ambitious and your students have seen integral calculus, you can introduce the Fourier transform and show the equivalent of a basis in function (Hilbert) space. An excellent reference is Korner's Fourier Analysis. It has many examples of applications: lots of physics stuff, how you can use fourier analysis to estimate the age of the earth, and how it has applications all over mathematics.
My real recommendation is to take some books out of your local library and skim them yourself for topics to present in class. Pick interesting stuff that will engage students with the limitless possibilities of mathematics.