I am a current undergraduate student and remember the feeling of wanting to know more before starting my degree. My university sent out a letter recommending that we all buy a certain analysis text before the start of the course, and that if we wanted to do some work before term that this was the book to read... Suffice it to say that the book was awful, and not even particularly useful once I had started at the university... a complete waste of my time and money.
Whilst this is slightly biased towards my own interests, if you are interested in combinatorics at all then I fully recommend 'Combinatorics and Graph Theory' by Harris, Hirst and Mossinghoff. The book is certainly readable for an enthusiastic school student, and really covers the basics of combinatorial enumeration and selected topics in graph theory. What's more the book is written with great clarity, and a good sense of humour. It starts at a basic level, but works up to material that there was not room to fit into a term long second year module.
Another text that I cant speak more highly of is 'Proofs from the Book' by Aigner and Ziegler. This is a compendium of interesting and simple proofs taken from many areas of mathematics. The title refers to a line of Paul Erdos. To Erdos, a book proof was a beautiful, elegant solution to a problem... so that is exactly what you find in here. Not all of it will be accessible immediately, but a lot will, and by the time you finish your first year you should be equiped to understand all of it.
Finally, from a softer approach: Martin Gardner's columns from Scientific American have been collected into several volumes which are readily available. Gardner often wrote about pure mathematics but also sometimes about philosophy of science, or physics. The articles are written so that the reader needs no technical knowledge, but I still read through the books just to find out a little bit about something that I haven't studied. Even if it is something that I do know about, reading Gardner is still a great exercise in seeing how good teachers really work: and there can be no doubt that Gardner was an excellent educator.
Best of luck.