I second Conceptual Mathematics for people who do not have mathematical maturity, but I would suggest Category Theory in Context by Emily Riehl as an intro for mathematicians (to be intended broadly). The textbook is rather gentle and provides lots of examples which are not heavy on theory (unlike Categories for the Working Mathematician), plus a tour of all of the main theorems (unlike Simmons), either proven or as doable exercises. It also introduces different approaches to achieving some results by presenting a graphical language to talk about natural transformations, which is useful when moving on to Enriched Category Theory. Riehl's book is available online for free on her website.

Riehl herself suggests Leinster's Basic Category Theory for people who want a lighter and easier introduction, but be aware that it covers way fewer topics and I can not vouch for it as I have not read it. It is available for free on arXiv.

A suggestion for more advanced readers who already have a high level understanding of the main theory and want to dive deeper would be Borceux's series Handbook of Categorical Algebra, which has both more breadth and details, covering even Topos Theory, Enriched Category Theory and so on. Many statements left by Riehl as exercises do have proofs there. You can find the first volume here.

Regarding ∞-Category Theory, people should approach it through Cisinski's Higher Categories and Homotopical Algebra, which is the most accessible complete introduction right now and also the only one developing a full theory of localizations. You can download the latest version for free on Cisinski's website. This book gives a less terse exposition than Lurie's Higher Topos Theory (the original complete reference) and many more proofs and details than all of the other introductions, like Land's Introduction to ∞-Categories.

Updated to introduce comparisons with other materials.