Skip abstract algebra, topology and analysis. If you find yourself in the same room as a number theory book, walk away slowly without making eye contact lest it cast a spell on you.
Sure, skip number theory. Like I said, you could swap that out.
One can learn the necessary topology, analysis (etc.) in the relevant places (and the relevant depths) that they come up.
1. The context is knowing more math than someone who has an undergraduate degree in it,
2. Abstract algebra is part of such a degree, and contributes significantly to overall mathematical maturity, and
3. You can avoid some subjects in the short term, but in the long term you can’t progress further without a reasonable mastery of algebra and analysis.
Probability theory and linear algebra are heavily used in data science. You won’t be as competitive a candidate for a job if you don’t have a firm grasp of both subjects. At a certain point, linear algebra ceases to be distinct from abstract algebra, and those exercises you were doing become applicable to real world results.
Read through, and do all the exercises in, one textbook each for:
1. Calculus
2. Linear Algebra
3. Abstract Algebra
4. Analysis
5. Topology
6. Probability Theory
7. Number Theory
...more or less in that order. Make sure your calculus book covers single variable and multivariable calculus. Supplement with applied mathematical statistics. Do that, and you have the equivalent of a mathematics undergrad (as far as relevant courses are concerned).
You could even do this with something like UIllinois’ NetMath program, or some courses on Coursera. You can swap out Number Theory for Complex Analysis or deeper Probability Theory and it’d be more relevant.