When you walk on stage, there will usually be some welcoming applause. Acknowledge that with a smile to the audience as you walk on, stopping to take a small bow once you near the piano. You don't want this bow to be too much beyond perfunctory, as you haven't done anything yet! Enter the bench to get seated from the front (audience) side, not the other side. Adjust the position of the bench. Do not play a note until everyone has settled down and you have silence. Until that happens just look down at your hands, and be thinking of the start of the first piece.
If your're giving a solo recital (as I have), there should always be a printed program, so the only announcement you will have to make is the title and composer for the encore. More on that to answer your question in a moment. If you are participating in a student recital, there may or may not be a program. If not, then you can announce your piece from the bench, projecting your voice to the back row so everyone can hear.
In a solo recital, you should only get up to bow at certain logical points. That would be the case after a Beethoven sonata, for example. But if you have a group of three short Chopin preludes as part of your program, it's best to give a smile and nod while seated on the bench to acknowledge the audience's applause after the first two, then to get up and bow after the third having finished the entire group. If there will be an intermission, you'll absolutely need to bow after the last piece of the first half. So you'll need to think about your program and plan ahead on this.
The bow I like is what I refer to as the "Paderewski Bow". Get up from the bench, take a couple of steps until you reach the end of the keyboard. Firmly take hold of the piano case (where it curves down in front of the key block) with your left hand, and do a fairly deep bow. If the applause continues take another bow, and even a third if necessary.
Have two encores prepared if possible. If your solo recital has gone well, the audience will let you know it at the end with their appreciative, sustained applause. There is no mistaking it. First, give them a substantive piece. If they demand more after that, play a much briefer witty or lush piece and escape. A great encore piece in the latter category, and one that nobody ever thinks of, is Rachmaninoff's "Lilacs".
Good luck in your performance!