First of all, you need to avoid using the words “playwright” and “paid” in the same sentence, because they rarely go hand in hand. When you’ve written your first play, chances are you will not be offered money to produce your show… You’ve just got to look on the bright side: Your baby is getting produced! But even if you are lucky enough to be compensated, yes, the deal at that point would just be between you and the theatre (and that’s with an -re, buddy).
But let’s say the play gets terrific reviews, and another theatre gets a whiff. (Or maybe your play gets good reviews and you send the manuscript to another theatre yourself, not a bad move as long as they’re not “big” enough to have time to care). Great! Two productions, now we’re rolling!
So at this point you may want to see if you can get published. This works the same way as getting a book published works… You send your manuscript, along with press clippings (if requested) to play publishers. For a beginner such as yourself I recommend Playscripts Inc., Bakers Plays, Samuel French Inc., and Dramatic Publishing, for starters.
Let’s say they publish you. Great, now you can sit back and collect royalty checks. They’ll print out your scripts for you, and whenever someone wants to do your play they call the publisher, pay the publisher (who in turn pays you) and you don’t have to do a thing. Of course I’m half-joking… You can still promote your work, but at least now it’s like you have an “agent” of sorts.
Like with all arts… music, painting, theatre, etc., don’t expect to get paid at all or a whole hell of a lot at first, but the name of the game is getting your work out there to generate a buzz, which enables you to get produced more often and hence, published.
And by the way, you’re right about not making it to Broadway. The only straight plays (as opposed to musicals, wiseguy) that make it to Broadway are written by gargantuan names in the business… Neil Simon, Richard Greenberg, Edward Albee, etc… And those rarely last more than 6 months… everything else is a musical.
Better to set your sights on the wealth of theatre opportunities in Los Angeles, Chicago, Florida, and Texas, to name a few. Good luck!