Can anyone recommend good books/tutorials on set theory/logic with simple explanations for a person with no math background (nothing beyond arithmetic and basic algebra back in school)?
Daniel Velleman's How to Prove It (2nd Edition).
It's basically written for exactly your purpose, and it's quite good.
That's a difficult question to answer well without knowing more about why you're interested or what you hope to accomplish. Logic and set theory are both very broad areas, though they don't look like it from the outside. My suggestion would be something like Set Theory and Logic by Robert Stoll. This is available as a Dover reprint or a Kindle book, for about $15. He discusses sets first, and only goes on to logic after that. The book has examples and exercises. I like his approach because if you're interested in sets you don't want to get bogged down in logic, until it becomes necessary. The Amazon link is http://amzn.com/0486638294. -- That's what I'd suggest, but I'm sure there are others, maybe better.
Keith Devlin's Mathematical Thinking is, in my opinion, an excellent book. It was written for his Coursera course named "Introduction to Mathematical Thinking", which I think is what you want. It is quite a small book, but he provides references for continuing the studies in basic areas of mathematics.
Such a book exists and is very reader friendly; it is Introduction To Modern Mathematics by Helena Rasiowa. This book along with How To Solve It by Polya are great for anyone starting out in mathematics.