Okay. Anthologies are a pretty easy market to crack. They aren't great for getting awards, but the over-all income and readership can be better.
Here's how to get into an invitation-only anthology:
Find an anthology. Note who edited it. Write to that person (with an SASE) asking if he/she is planning to do any more anthologies, and if so, may you submit? Odds are the reply will come with an invite (if the editor is, in fact, planning another anthology (and most of them are.))
Then: when the time comes, make sure your story is the best one in the anthology. This isn't too tough either.
The story by the big-name pro that the editor had to get in order to sell the anthology to the publisher won't be his best work. It may be a trunk story, or it may be something that he knocked off over the weekend. The stories by the editor's friends aren't going to be very good, but he can't turn them down after he asked them to write 'em. So that leaves you. You write something dazzling, and you either get slotted into the lead or the last position in the book (both good places to be). There you are, published in an anthology!
Now as to magazines:
Abyss & Apex, Analog Science Fiction & Fact, Apex Digest, Asimov's Science Fiction, Aurealis, Boys' Life, Brew City Magazine, Cosmos Magazine, Cricket, Doorways Magazine, Fantasy & Science Fiction (F&SF), Futurismic, Grantville Gazette, Interzone, On Spec, On The Premises, Orson Scott Card's Intergalactic Medicine Show, Paradox, Serpentarius Magazine, Strange Horizons, Transmitter, Writers of the Future Contest, Wrong World,
That should keep you busy for a while. Remember, VERY IMPORTANT, get and read the guidelines and several sample copies before you submit anything to any market. You do not get a prize for submitting material that isn't within shouting range of what they publish. Each of them has a style. It's up to you, young Jedi, to understand the markets.