It really depends which direction in mathematics you want to go. Even as a math major, I didn't really understand how vast it was until I got into abstract math.
My favorite way to learn is browse Amazon for "Dover Books on Mathematics." They are generally had for a penny + shipping if you don't mind buying used.
A good intro into modern mathematics: https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247
Concepts of Modern Mathematics by Ian Stewart.
“According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.”
How about selected chapters from Stewart's Concepts of Modern Mathematics? It has a pretty wide range of jumping off points and is a relatively affordable Dover book. You could go into more or lesser detail on these topics based on the students' backgrounds.
Another idea would be to focus on foundations like set theory, logic, construction/progression of number systems from ℕ -> ℤ -> ℚ -> ℝ -> ℂ , and then maybe move into some philosophy of math. There could be some fun and accessible class discussion, such as having them argue for or against Platonism. [Edit: You could throw in some Smullyan puzzle book stuff for the logic portion of this for further entertainment value.]
I enjoyed Concepts of Modern Mathematics when I was in high school. It might be a little basic and it's a bit uneven in places. But it's a really good lay account of the basic notions in "modern" mathematics. It doesn't really mention so much the various fields. For that, surfing Wikipedia is hard to beat.
Take a freshman proofs class (or any "Fundamentals of Math" course) so you'll get a solid grasp on proof techniques, then pick up this book to get a neat overview of some interesting topics. From there it's entirely up to your desire to learn more.
This book will introduce various fields of math in an entertaining and accessible manner, aimed at someone around your level:
https://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247/
Hmmm. Try:
Whatever Happened To New Math?
And:
A history of the "new math" movement in the United States by Robert W. Hayden
I did SMP mathematics for O-level (age 16 exams) and A-level (age 18 exams) which was sort of similar thing in the UK. As it happened, going into Computer Science, a lot of the abstract stuff maths we covered turned out to be quite useful for me and put me ahead of other people on my course. It's also helping me n years later now I'm working on a mathematics degree.
There's an interesting book written by Professor Ian Stewart:
Concepts of Modern Mathematics
He describes the original genesis of the book in the preface as being a course he created for parents to try and help them to come to terms with "New Math" as it was being taught in schools at the time. If you're looking to learn more about the concepts then that might be the sort of thing you are looking for.
Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos