For a modern popular math book I would recommend The Man Who Loved Only Numbers (by Paul Hoffman) about the life Paul Erdős. Is a great intro to the life of a working mathematician and a bit about proof.

The classic text is How To Solve It (by Georg Pólya). A bit antiquated but still good.

Beyond that: find an introductory text book to a subject and start trying the worked examples!

You'd recommend Polya over "How to Prove It" by Velleman? Polya's seems more about general problem-solving techniques, versus proofs.

Thanks for the recommendations!

Isn't that a bit misleading because Erdos was a famously unusual mathematician personality/lifestyle? I think focusing on Erdos as an intro fetishizes mathematicians/mathematics.

Good point but the book is very clear about how unusual Erdős was and talks about the lives of more normal mathematicians he worked with.