TURN OFF YOUR PHONES BEFORE THE LIGHTS GO DOWN ("silence" does not do it: alarms override; some phones have minds of their own)...you will be noisily startled and mortified if your phone goes off during the legendary actor's climactic monologue at the end of the first act...and believe me, that's when it will go off...
On a related note, do not talk during while the house lights are down and/or the orchestra has started playing. Remember that this performance is a once-in-a-lifetime experience, and if you talk to your neighbor (even in a whisper), you're interrupting the experience for someone else (and your neighbor may not want to be interrupted either). Unlike a movie, people cannot go back and watch it again. Intermission will be along soon enough; or, if the show does not have an intermission, it is generally under 2 hours in length.
Assume the artists who created the show have a reason behind everything that happens on stage~when the lights go down and/or the orchestra begins to play and/or you notice activity on stage (even if it seems casual), quiet down, and start paying attention~then you won't miss something interesting or important, and your neighbors will truly appreciate you! Even before the show starts, I suggest taking a minute to notice details of the set or any visible part of the stage; there are often clever references that take a keen eye to recognize.
Get to the theater with at least 15-20 minutes to spare so that you can use the restroom (ladies, there will be a loooong line), check out the merchandise kiosk and enjoy the physical space of the theater. Many theaters are really quite lovely -- be sure to look up :)!
If you are in New York City, keep in mind that the Theater District is jammed before and after performance times (Wed, Sat & Sun afternoons and every evening except Sundays & Mondays), and even walking a few blocks can take time and patience. If you're in a group, keep close together, but don't block the pedestrian traffic to do it. Just make sure that everyone in your party knows the address to the theater, and, if possible, has their ticket in hand. Speaking of hands, take small children by them.
If you're sitting in the middle of a row, get to your seat a minimum of 10 minutes before curtain, and make sure you're ready to stay put once you've gone into the row.
At intermission take a few minutes to look around the theater; enjoy the scenery but also look to see who else came to see the show...it's interesting to observe who the audience for that show is that night. And you might see someone you recognize! I've seen famous folks, but also run into friends that way too :).
Be attentive at intermission as to the comings and goings in your row. Try to coordinate your ins and outs with others you notice leaving so that someone staying put does not have to get up and down numerous times.
If you want to go to the stage door after the show (trust me, you probably do :)), see if you can find out where the door is before you enter the theater; then you can head right there after the show.
Buy any souvenirs you want before the show or at intermission so that you can head right to the stage door afterwards.
If you're at the show by yourself, don't be shy about chatting with your seat neighbors (if you feel like it); you may meet someone really interesting.
Read the Playbill for interesting information about the show and to figure out why that supporting actor looks so familiar to you!
Dress in layers so that you won't be uncomfortably warm or cool during the show (and you can always use any extra layers as booster cushions if the person in front of you is in your way :)).
Speaking of booster cushions, many, if not most, theaters have them available. They are generally intended for children, but if you are particularly height-challenged, and sitting behind someone who is on the extremely tall side (or who inexplicably leaves their hat, glasses, etc. on top of their head during the show), don't be shy about asking for a cushion from an usher.
Speaking of hats, remove them. Assume the view of the person sitting behind you is going to be obstructed. No one cares what your hair/head looks like without the hat, especially in the dark. You can always put it back on at intermission!
The ushers are not kidding when they tell you not to take photos inside the theater, not to place coats or other things on the ledge of the mezzanine/balcony, to turn off your phones/tablets, etc. They will find you and publicly admonish you. It's not pretty.
TURN YOUR CELLPHONES ALL THE WAY OFF! Yes, it bears repeating. Every single show I was in this weekend, from large to small, featured someone's cell phone ringing despite many, many reminders before the curtains rose. It is incredibly disruptive. Don't be that person.