This book is fantastic and should get you pretty far: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

About 6 months before starting my upper division applied maths coursework, I picked up a copy of Baby Rudin, slogged through chapter 1-2, decided it was an inefficient use of time, and contented myself to wait to learn from my future courses' lecturers. Now I'm about 6 months into my upper division coursework and the most important lesson I've learned is that the only way to learn the concepts is to slog through them on your own. The pace is usually so slow you feel like you're not actually learning anything. But every once-in-a-while a couple pieces fall into place, whatever puzzle you're looking at gets a little clearer, and you realize you're growing and making progress.

Anyways, my point is that I bet you can do it, you should try, and it might feel hopeless at first. Don't be surprised or discouraged if it takes you something like an hour per page (or more) to work through a textbook. The good news is that's probably all you really need to do: work through it.

I wouldn't recommend Rudin for a first exposure to Analysis. You will undoubtably suffer unfairly if you look at Rudin and aren't in the context of a classroom and/or you lack the mentorship to have your proofs checked. It's one of those books highly recommended but unuseful for the absolute beginner. I heard better things about Spivak for the newbie.

But I would Terrance Tao's notes on analysis: http://www.math.ucla.edu/%7Etao/resource/general/131ah.1.03w...

Before you start with Tao go through all the calculus series on Khan academy & get an introductory book on proof writing.

It obviously depends on background and goals. People who come to programming with only HS math would probably benefit most from studying (not just reading) an introductory discrete math text such as Biggs or Rosen. For calculus, linear algebra and so on one could look at MIT Opencourseware. For most people, actually learning the material requires a significant time investment, and one of the hardest things to fake is a knowledge of math.

> That seems about impossible to me, it's a subject where you'd really need study groups and office hours.

I actually never really liked study groups so to me studying math is the same as studying any other subject, but everyone is different. Considering you do well with study groups it seems wise to first try to find others who also want to study so you guys can study together.

It's slow, but working through good books is a valid path. It's a huge boost if know a math grad student to give you a bail out / explanation when the exercises get difficult (or you can't figure out why the book puts focus on specific topics).