Set Theory, Number Theory, and Logic before any other math I am trying to teach myself set theory, number theory and logic before I engage in mathematics. I have been able to get through Calculus but I think that it was just by repeating different types of problems (i.e., memorization). I really want to understand the mathematics in a pure format as I think the applied format will come from that easier. I am not interested in being a mathematician but will need a lot of math in what I will be studying and thought it would be good to have a solid foundation in mathematics. Would these  subjects be a good place to start before trying geometry/precalculus and the rest of the traditional mathematics sequence?
I am hoping the logic based approach will help me be more creative when I actually have to conduct some type of research that requires a great level of innovativeness.
Thank you
 A: Personally I am taking Keith Devlin's Introduction to Mathematical Thinking on Coursera right now and it is amazing. It changed how I viewed mathematics. 
He wrote this really awesome short essay here:
http://spark-public.s3.amazonaws.com/maththink/readings/Background_Reading.pdf
Read that essay. It might change your life. Seriously. The key highlight for me anyway was that mathematics these days is defined as "the science of patterns" and is more about seeing patterns and finding truth. When you phrase it that way it suddenly has much more appeal for people like me who are obsessed with finding patterns by reading history and studying sociology. Also it got me into theoretical CS instead of just hacking random things together. 
This pretty much blew me away because I only took up to Linear Algebra in university and I thought math was this stupid calculation based thing. 
Proofs are insanely awesome. They're hard and they make you think. And they're beautiful.
https://class.coursera.org/maththink-004
You can sign up for the class here and go through the material at your own pace. I think it's a good idea for someone in your position because it's a survey class. I'm sure you can ask around in the forums for ways to go deeper if you're interested. 
A: I don't agree with you, when you say for Calculus that it has to do with "repeating different types of problems (i.e., memorization)". Start reading (theoretic) books about analysis, and you'll see the deepness of that!
You also can read something about relations, vector algebra and linear algebra.
A: Some familiarity with logic and set theory is required for studying calculus also,  but as far as a systematic study of these topics are concerned, you should do it if you are really interested in it. You won't be needing very advanced knowledge of logic for studying topics like Abstract Algebra, Linear Algebra, Analysis etc. However it is a beautiful subject in itself and studying it will definitely prove beneficial. Similar reasonings hold for Set Theory with the exception that there are certain topics (like Zorn's Lemma) which will be needed in Algebra or Analysis, so why not read it properly.
Studying Number Theory will give you an exposure to $proofs$ and different ways of proving things. Moreover, its very easy to check theorems/lemmas with examples in Number Theory so one gets a $feel$ for the subject. Neither it is a prerequisite nor necesaary to study calculus. So if your main aim is to study calculus then it can be done with some knowledge of set theory and proof-writing. But if you want to learn about different areas then these are definitely worth looking.   
