"Publishing" is done in regular book form of the type, "101 skits for you and your family to put on" or "Ten short plays teaching children about (fill in the blank).
As for what you do with a real play.
Aye, there is the rub.
You have to find a Backer, someone to finance, produce, put on, your play. In my case the "backers" come to me. The heard I wrote a play for Parents without Partners, or a boy scout troup, or a church organization. They basically tell me what the theme is, what it is for, say to raise money, or get a point across, or something, and what their resources are. I give it my best shot, for free, and usually everyone is happy and has fun. This happens as often as 3 times a year to as little as once every 3 years.
But a real play.
You are trying for success, money, good reviews, prestige...For all of that you need real actors, paid a salary, directed by someone who knows what they are doing, performed on a real stage.
A successful play can make big money for everyone. Most fold within a week, many after the first night. If you write something to equal Cats, you are a hero.
Unless you are indepenantly wealthy, or marry into money, such as George Bernard Shaw, you will have to rely on backers. These people are not easy to find. Few people are interested. The ones who do it often have successful playwrights chasing them down.
In other words in order to attact them you need to have some successes under your belt.
My suggestion is to attach yourself to every little theater group you can come across in your area. Learn everything you can, not just about the writing, but about the audience, the actors, the director. Write something and see how the audience is affected by it.
Your first effort is liable to be along the lines of adapting something well known, like Taming of the Shrew, for the needs, abilities, finances, settings, and the number of available actors, of a local group.
Sound dumb? It is in a way. But if the original play calls for 5 women, and there are only two in the troupe, you must do something. What if the two you have are both of the wrong age to play Juliet convincingly?
I don't think this is the answer you wanted, but I hope it helps. And times change. I haven't updated my info base on this subject in some time. I'll listen to anyone with recent experiences who wishes to comment.