A water bottle is necessary, and I've never been to a studio that gives them away. I strongly recommend you bring a change of clothes, especially if you're going to a hot yoga class, unless you're down with going home in wet, heavy, smelly clothes. You might be able to get away with just a shirt change if it's not a heated class, but that depends on how easily you sweat.
Silence your phone
Not every studio is big enough to have a separate room for your belongings. Even if there's a locker room, obnoxious ringtones can often be heard through the walls since classes are either quiet or have relaxing music playing.
Take off your shoes
It's a barefoot space. If that gives you the willies, well, what, you never take your shoes off?