So what do you do after you finish the first draft of your story? If you answered "revise it!", you would be correct. However, this process doesn't just involve you, your story should also be read by others. Having your story read by others is a great way to not only detect the grammar mistakes you didn't see, but also another way to get ideas for your story.
Usually everyone reads their story out loud since the story sounds different when you hear it rather than read it. You can follow along and make any other changes you haven't already made. After the person is done, you state what you liked/disliked about the story. By dislike, I don't mean "That was a really crappy story," I mean "I didn't really understand why you put this there" or something like that.
How you pick your group all depends on what your needs as a writer are. Do you need a group where everyone isn't very experienced when it comes to writing stories? Do you need an experienced group? A group with a mixture of both? These are just some questions to consider.
Oh, and here's a site with some tips on getting started with your writing group.