I second Siming Tu's recommendation for E-W book.
It is a well balanced book (regarding theory vs applications), it has nice appendix contains relevant theory from functional analysis, and it contains a nice selection of subjects (although not addressing entropy, which one might say is a very big problem).

I think that overall, Petersen's book is good as well, maybe not as streamlined as one might expect, but still very through.

So apart from this, which are "standard references", and maybe also Walters' book (which is kind of dated, and the last chapters are biased towards entropy theory of continuous maps over compact spaces), there are few references which are good for specific subjects and maybe not as a whole standard reference book.

Dan Rudolph have a very nice book called - "Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces", it is one of the most accurate books in the technical level (Lebesgue spaces, ergodic decomposition), plus it have nice treatments of the theory of joinings and entropy. He also actually proves Ornstein's theorem, kind of a rare thing.

Another nice option is the classic book by Furstenberg - "Recurrence in ergodic theory and combinatorial number theory" (Princeton).
It is an extremely suitable book for students I think (because Furstenberg is a great teacher and lecturer), and it shows part of the motivations towards the modern development of ergodic theory, and it shows also topics in topological dynamics, which other books omit.
Nevertheless, it does not as extensive as E-W or Petersen on the ergodic theoretic part, but it definitely worth your time after you got the hang of the basics.

The last option I have in mind is Shmuel (Eli) Glasner's book - "Ergodic Theory via Joinings" (AMS).
This is a very extensive book, but it is kind of deep, and in my opinion, doesn't suitable fro students (although he for example discuss the general notion of ergodic group action, besides Z or R actions).