How to go from beginner to expert in mathematics?

I think the "How to become a pure mathematician" website is what you're looking for. It offers a well structured approach from very basic mathematics all the way up to graduate level, with links to useful resources and books.

Also, any university curriculum can be useful to see in what order university students study different subjects. An example is Cambridge (the famous mathematical tripos), where they have pdf's outlining every part of the course:

• Part IA
• Part IB
• Part II
• Part III

You will also find past exam papers and a bunch of other resources. Very useful.

This site as well as wolfram|alpha are both excellent resources for teaching yourself math!

I'd recommend looking at the KhanAcademy: it's online, free to sign-up and participate, progresses in level of difficulty, and is self-paced. It is good for reviewing the "basics" and takes you through calculus and a bit beyond.

See also this site for mathematics-related on-line video tutorials - literally hundreds of them!

Another helpful resource for learning algebra-precalculus, calculus, linear algebra and differential equations Paul's Online Math Notes and tutorials.

Then perhaps you'd like to explore MIT's Open Courseware - Mathematics for access to classes and topics of interest in more advanced topics. From there, you'll find a list of classes with resources available, and will also learn which texts are used for the available classes. Often, course notes, videos of lectures, and exams are available to assist learners, all free of charge.

For learning how to read and write proofs, including proof by induction, a wonderful text to read and work through is How to Prove It: A Structured Approach by Daniel Velleman.

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You might want to look at the following mathematics classification guides to see the scope of courses and areas that are available:

$\bullet$ The Mathematical Atlas

$\bullet$ American Mathematical Society's Mathematics Subject Classification

$\bullet$ Engineering and Physical Sciences Research Council - Mathematical sciences

$\bullet$ Mathematics arXiv - Categories within Mathematics

This makes answering your question difficult because it is not entirely clear what your end goals are.

You certainly should learn proof techniques using such books as

• Problem-Solving Strategies, Engel (somewhat high-level)
• How to Solve It, Polya
• The Art and Craft of Problem Solving, Zeitz

From all of the above, you should see if you can refine your goals and the wonderful MSE Community can provide more guidance.

Regards