What's the best way to learn mathematics outside of formal education?

Take university classes - it's the same material, you just won't get any academic credits. MIT Opencourseware has Single Variable Calculus (18.01), Multivariable Calculus (18.02) and Linear Algebra (18.06). That's plenty of material to get started with. Watch the lectures, follow along with the textbook, do the exercises.

If you're concerned that you don't have some of the prerequisites like high school algebra or trigonometry, Khan Academy have a pre-calculus course - start there.

I switched to data science from an unrelated field and had to do a lot self-study like this to catch up on my math.

It's a hard subject to teach yourself. I was in a similar situation to you, and after starting with various books I ultimately ended up enrolling with the Open University for a BSc in Mathematics.

Before that though, the most useful thing I did was to get the OU course books for their introductory courses (MST124 and MST125). They're written for self-study, and they're extremely comprehensive. These books filled in all the gaps and ironed out all the kinks in my fundamental mathematical knowledge. Things that used to baffle me make perfect sense now, like why the rules for non-integer exponents work the way they do.

Another book I'd recommend for someone in this situation is Measurement by Paul Lockhart. It won't exactly teach you maths, but it'll teach you what maths is, which is an important thing that most people are missing.

The problem with self-learning is lack of feedback. You read something, you think you understand, but you might not, and you don't even know it. I once read Capablanca's "Chess Fundamentals" book, and I felt like I made a quantum leap in my chess understanding. One day later a friend of mine beat me 5 times in a row, so I learned the super-valuable lesson: self-assessment is not always very reliable.

Given that, how can you get feedback when you learn math on your own? I suppose there are various ways, but you can't go very wrong with Khan Academy. I just went to their website and practiced limits a bit [1]. I've known limits for 30 years, in and out, but I still found these 3-4 minutes practicing their exercises to be quite fun. I even got some stars and badges :)

I'm not affiliated with them in any way. The site just looks good to me. Give it a try.

[1] https://www.khanacademy.org/math/calculus-1/cs1-limits-and-c...

My method has been to audit a university class, do lots of practice problems, and find a support group of people to regularly work with

If youve taken say, trig, but are rusty, dont bog down on trying to relearn upfront. If it comes up when learning something else, then relearn whats necessary. The problem with this is if you're in a scheduled class where you might get behind. The benefit is that when you go on to upper level math, its a necessary skill to relearn some things quickly

I know someone who wanted to build up a solid mathematical foundation after years of considering themself to be bad at it, and ended up loving Spivak's book on calculus. (Another well-written book on calculus, in a very different style, is the classic book "Calculus Made Easy" - this book is old enough to be available on Project Gutenberg.)

You can learn linear algebra without learning calculus, but to really care about linear algebra you have to know why it's useful. The main lesson from (multivariable) calculus is that you should approximate functions by linear functions whenever possible - so the basis of almost everything else in math is a deep understanding of linear functions (and the occasional quadratic when you need that little bit of extra oomph). Once you manage to convince yourself to care about linear functions, the subject magically becomes easier to learn. I don't have any especially great textbook recommendations - perhaps Strang's book?

YouTube has a tonne of great presentations on different subjects like these, otherwise you could try the Khan Academy.

You might even find some decent text books with exercises online but if you prefer to watch etc, YT is great.

However, if you want to be able to actually use and apply it to a decent standard, you are probably going to need something more formal by way of teaching.

Time and effort.

True story: I college I had an upper level math course. On day-one the prof comes in and says (paraphrasing) "You've got the book. Read it. If there are questions, I'll help. Otherwise, you're on your own."

We left that class pretty angry. He's not going to teach us? Etc.

Long to short, it was one of the best things that ever happened to me. Up til that point, I was always taught such things (i.e., math and technical material). From that class forward I had the attitude, confidence and mindset to teach myself. Let's just say the number of technical lectures I attended the rest of my college career was cut in half.

Dive in. Find the medium your prefer (e.g., video, books, etc.) and keep going :) Ideally, find someone else who shares your curiosity so you have share the highs and lows.

All of the comments here are valuable advice. Perhaps the only things to add are that learning math is cumulative, and learning math is like learning a new language. In other words, the more difficult concepts are usually built on other fundamental concepts. Be sure to build your foundation carefully, and if you have difficulty with any concept, look for your holes in understanding and don’t be deterred or discouraged from going back and relearning the fundamentals. Find your process and stick with it. Good luck.

Mathematics not only allows us to calculate complex equations, but also helps us in everyday life. Learning is not always easy, and someone is not interested, especially in the school version, so it is better to find exactly the format that suits you best. There are so many different platforms to explore now, such as Udemy, edX, YouTube. Perhaps you can find something interesting there.

Wouldn't it be Magic to interact with Siri, verbally and visually, on the iPad to get to the gist of what you need in 15-30 minutes and continue on your quest?

Do the problem sets.

https://www.youtube.com/c/3blue1brown

Books from Dover are really cheap! You can also find lots of old, cheap, high quality math textbooks on Amazon and ThriftBooks.