My Background: Current Undergraduate in CS and I recently added Mathematics
The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on mathematical thinking and proof techniques before/during/after reading any other dense math book.
Like you, I realize the value of having a mathematical mindset and want to have a deep understanding. When I added math as a major, I had a very hard time jumping from computational courses (typical math courses, geared towards any major) to theoretical and conceptual courses (proof-based courses that use all the fun and interesting math books everyone has linked here). These books helped:
<3 this is a great book, obvi since its george polya
“How to Read and Do Proofs: An Introduction to Mathematical Thought Processes”
For easing into higher maths I recommend this one on proofs: https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat....
Also if you're concerned about quality the MAA reviews math books and publishes a basic library list for undergrad maths: https://www.maa.org/press/maa-reviews.
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