How do I learn subject $X$ in Mathematics?
The best option: ask a knowledgeable Mathematician! Mathematics is a fundamentally human activity, and abstractions are much easier to communicate in person. Doing a course or guided reading with someone like a Professor or a graduate student is perhaps the ideal option here.
The next-best option: find a good book and/or course materials
Find a motivation
Math is technical, and it’s hard to slog through details without a motivating factor. Find a non-trivial result that sounds interesting, and investigate how the subject you’re studying supports or leads to that result.
For example, if you wanted to learn Galois Theory, it’s worth some time reading up on things like the general insolubility of quintic equations
Find a good textbook
You’ll probably want to start with a good textbook – I’d first recommend checking my own resources page to see if there’s anything listed there.
A second recommendation is to Google something like (for example)
reference request algebra site:math.stackexchange.com, and to look through results specifically from Math StackExchange and Math Overflow. Reference requests on these sites are often turned into community wikis, and so the number of upvotes can serve as a good first-approximation of which books are worthwhile.
Skim Once you obtain a book, skim a few chapters to see if the writing style clicks with you. Also read the author’s introduction in full, as authors often lay out high-level motivations and goals in introductions.
Useful trick: Find a book for which the topic your book covers is a prerequisite. Math books often has appendices which provide brief overviews of prerequisitive material. For example, if you wanted to learn point-set Topology, you can read the appendix in Lee’s Manifolds book – short, concise, and best of all, gives you an idea of where the topic is actually used!
(Optional) Find lecture notes or other material from courses that use your book, if possible. This can be done with a Google query like (e.g.)
inurl:edu dummit and foote syllabus. These often help highlight the more important or major topics.
Read, take notes, solve problems, and consult other resources whenever necessary!
What is subject $X$ used for / what are the real-world applications of subject $X$?
I defer to this excellent answer on MO: Real-world applications of Mathematics by Arxiv subject area.
What are some fun activities or events for a Math club?
- Lightning Talks
- Board Game Nights
- Celebrate Math-related holidays: $\pi$ day, Fibonacci day, or make one up!
- Math Movies
- Problem-solving sessions or competition prep for things like the Putnam.