You look for publishers who publish “that kind of thing”, whatever it is. You send them what you’ve done (a letter asking if they’d like to see a whole manuscript or a few chapters and an outline will always be welcome. And stamped self-addressed envelopes help keep the wheels turning.)
Sooner or later, if you don’t give up and you have some measurable amount of ability or talent or luck, you get published. But for people who don’t know where to begin, let me offer a few suggestions:
Meet editors. If you’re into SF, Horror or Fantasy, go to the kinds of SF, Horror or Fantasy conventions that editors go to (mainly the big ones – look for words like WORLD or NATIONAL in the title). Same goes for Romance or Crime. Join associations – SFWA or HWA or the Romance Writers of America or The Society of Authors. Most organisations like that have an associate membership for people who wouldn’t qualify for a full membership.
Even if you haven’t met any editors, send your stuff out.
The “slush pile” of unsolicited manuscripts is not always a bad thing – publishers take enormous pleasure in finding authors from the slush pile (Iain Banks and Storm Constantine are both writers who simply sent out manuscripts to publishers), although it occurs rarely enough that it has to be a special thing when it happens.
If you write short stories, don’t worry about agents, just find places that might print the stories and get them out there. If you write novels, I think it’s six of one, half a dozen of the other. I’d written and published 3 books before I decided it was time to get an agent.
Writers groups can be good and they can be bad. Depends on the people in them, and what they’re in them for.
On the whole, anything that gets you writing and keeps you writing is a good thing. Anything that stops you writing is a bad thing. If you find your writers group stopping you from writing, then drop it.
The other thing I’d suggest is Use The Web.
Use it for anything you can – writers groups, feedback, networking, finding out how things work, getting published. It exists: take advantage of it.
Believe in yourself. Keep writing.