What does it take to get published?
Tenacity. It's easy to give up. Really. Any author can tell you that. You write the next Great American Novel (GAN) and there isn't a publisher out there who thinks your GAN is all that great. Of course, you only sent it to two publishers before you gave up, and started writing another GAN. (Or maybe that's just me.) Bottom line: If you want to actually be an author and not just a writer, then you have to work at it. Eventually someone will love your book as much as you do, and you'll get published.
Where can I find a publisher?
There are numerous books out there that list publishers, contests, agents, etc. The Novel and Short Story Writer's Market should be your best friend. It's published by Writer's Digest Books. They also publish the same book geared towards the Christian market. Do your homework! If you write a science fiction novel, don't submit it to a publisher who is only looking for historical romances. You're just wasting your time, and theirs. Go through the book and highlight publishers that are looking for your genre and submit to only them.
Do I need an agent?
That's totally your own decision. If you've never been published, it's going to be harder to find one, but not impossible. The Guide to Literary Agents (again by Writer's Digest Books) is a great start. And...do your homework! Most of them are looking for specific genres. If yours doesn't fit, don't bother. Just trust me on this one.
Self-Publishing. Yes or no?
Another thing that only you can answer. Self-publishing doesn't necessarily carry the same "stigma" that it used to when it was always called vanity press, but it does cost. In a perfect world, someone would be paying you to publish your book, not the other way around. If you can afford it and your goal is to see your name on the cover of a book, then it's worth considering. It might be a good way to get your foot in the door, so to speak, especially if your book sells well. The other side of that coin is, it's going to be hard to later sell that same novel to a traditional publisher if your book doesn't sell. You'll have a huge negative working against you. Check out the various self-publishing companies, and see which one fits you (and your budget) the best. And...be sure you Google/Bing/etc. the company for reviews, and/or check them out on the Better Business Bureau before you commit. Again, do your homework!