I've been a high school math teacher for most of my life, and I have deep frustrations with how removed from meaning math is presented to most students. Just because the teacher knows and states the possible relevance doesn't mean students should be expected to take the relevance at face value.
I was mostly focused on teaching algebra 1 classes, which is why I didn't use higher math all that often. But my understanding of higher math grounded my teaching of lower level concepts all the time, and I often spoke of higher level concepts with my students to help demystify math. My 8yo son loves math for now, and the moment school makes math meaningless to him I am planning to find some way to intervene.
a) You don’t do this full time.
b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.
c) You are skilled with programming/software in general.
In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.
Another thing you have going for yourself is - you can basically skip high school math and jump
right in for the good stuff.
I’d recommend getting great and cheap russian recap of mathematics up to 60s  and a modern coverage of the field in relatively light essay form .
Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.
Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.
For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.
I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.