My background is programming myself - I have a degree in bioinformatics and computational biology and spent most of my time doing things like text mining and such. Perl was/is my jam, and I spent more time in computer science classes than I like to think about.
But now being in education policy, I find that the core of my thinking has had to shift from programming to math.
Here are some of my tips, then:
- Find a topic you care about, with data you can find, and a question for which you want to know an answer. I found that having a focus like that gives my math-learning a real point (which I personally feel is a weakness of a lot of math education).
- If you like the programming, stepping through some of the data analyses done in R or Stata, I think, can really help you wrap your head around what's going on.
- Get a good book. There are lots out there, but I would say that which one you get (as well as which language you work on most) depends on what field you're in.
- Talk to people. I find it really hard to understand a statistics concept unless I can talk it out with people.
- Go down the rabbit hole. If you have a problem, start with the simplest solution you can think of. Then ask people why it might be wrong, and go from there.
I really think the biggest thing is to find a project you care about. I used to gloss over the math-y sections of things and skip right to programming, but once I found questions where it was important to me to get the right (or very close to right) answer, it made it much easier.