I'll recommend a couple of books from that thread:
I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.
Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.
Get that right away.
Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).
Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.
(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)
(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)
Concepts of Modern Mathematics by Ian Stewart
Numbers and Geometry by John Stillwell.
The Pleasures of Counting by T. W. Körner
Mathematics: A Very Short Introduction by Timothy Gowers
Numbers and Geometry by John Stillwell is very readable and full of very interesting math problems, covering a lot of topics that are accessible to anyone with a good high school education but often aren't covered even by an undergraduate math degree.