Some of you may want simply to learn set theory, rather than learn set theory in order to do category theory. Therefore, I list here a few of the most canonical texts used by set theorists---these book are all fantastic. None of them, however, is concerned with category theory at all.
Set Theory, by Thomas Jech. (3rd Millenium edition). This book is a standard graduate introduction to set theory, and covers all the elementary theory and more, including infinite combinatorics, forcing, independence, descriptive set theory, large cardinals and so on. It is used almost universally in any serious graduate introduction to set theory. Excellent text.
The Higher Infinite, by Akihiro Kanamori. This encyclopedic account of large cardinals is simply fantastic. It contains everything you wanted to know about essentially all the most well-known large cardinals. These cardinals form a very rich structure with a highly developed theory, including surprising connections even with the structure of sets of reals and much much more. Surely any talk of "universes" in category theory would be deeply informed by knowledge of the large cardinal hierarchy, and the far more nuanced and developed structure theory it provides for these concepts.
Set Theory: An Introduction to Independence Proofs, by Kenneth Kunen. This shorter book is an excellent companion to Jech's book, in that they have different approaches to many common problems. I always recommend my graduate students to play these books off against one another.
Of course, there are many others.
