Good idea to learn to read music! For the most part, most adult beginner books will explain that right up front with the starting information on what the staff is and how it relates to the keyboard. They will also discuss the most common types of notes (whole, half and quarter) and their time values. The staff represents the keyboard - where a note is at on the staff tells you which note to play on the keys. Which type of note they put there tells you how long to hold that note down, for how many counts.
If you want more explanation than some adult books provide (I've had experience with Alfred's Adult series - both their "standard" and their "all-in-one" series - with a few students and have been happy with them), you can pick up any childs beginner book and go through their explanation. Thompson's "Teaching Little Fingers to Play" is a fairly standard, stood the test of time, beginning book for kids; and Schaum's "green book" (can't remember the name of it off the top of my head - it's white with a wide - 4" or so -green bar down the left on the front, I think - I hope they haven't redesigned it - I bought several for my stock awhile back and haven't needed to buy any lately) is another "tried and true" recommendation as far as kids' lesson books.
I personally wouldn't recommend any books that start you off without the staff. Dive right in and start with the staff and middle C. I've only used these types of books (staff-less at the beginning) for kids who haven't learned to read yet. And then, only my own kids, because I'm not convinced that it's a good way to learn piano. Although I did have 1 student come to me from another teacher who had started her this way, and we got along just fine. But if "reading music" is your big goal, that will just waste your time I suspect.
Have fun! Reading music could open you up to a whole 'nuther world of music! From here, it's just a matter of learning "how" if you want to learn another instrument, since you would already know how to read music!