When you write literary fiction, literary non fiction, or poetry, it's highly unusual to have a book published without having first established a publication record. Start out by submitting your work to literary journals, entering reputable literary contests (beware of contests that require a substantial submission fee), and applying for writing scholarships and prizes. For a list of trusted and legitimate contests and awards, you can visit the award listings at Poets & Writers.
As you prepare to submit your creative work for publication,
Take the time to read the journals you're considering in order to gain a sense of the kinds of work they publish and whether your piece would be a good fit. Read journal content on line, request a sample copy, or obtain one through a local bookstore or library. (If you can afford a copy or a subscription, this is always nice: small presses and journals typically run on a shoe-string and rely on grants, donations, sales, and subscriptions to continue to operate.)
Read submission guidelines carefully. These include not only rules for formatting and submission but also information about when/whether the journal accepts unsolicited manuscripts.
Prepare a cover letter. This should be brief, and can include information such as your educational background and a list of your prior publications, prizes, awards, related scholarships, etc. Do not name drop, engage in a discussion of your creative work, or add personal details. Stick to a simple, straightforward format.
Be prepared to receive rejection slips. Each literary journal has its own editorial preferences, unique voice, style, and tone. Journals also receive hundreds (sometimes thousands) of unsolicited manuscripts, and obviously cannot publish more than a small fraction of the work they receive. Receiving a rejection slip does not necessarily mean that your writing is not good; other considerations, such as the fitness of your piece for the particular journal, editorial tastes and priorities, and the sheer number of submissions may be the factors that account for whether or not your work is accepted. If you feel your work has value, do not give up! Most well established writers have received reams of rejection notices over the course of their careers.
Some campus and local opportunities for publishing or showcasing your creative work:
Bricolage, the undergraduate literary arts journal
AU, the undergraduate speculative fiction journal
Blind Glass, the undergraduate poetry journal
Castalia, the UW's literary reading series
Write Away!, a UW writers group (must be logged in to Facebook to view their page)
Manic Mouth Congress, a campus poetry slam collective
Richard Hugo House Literary Arts Center, Capitol Hill
Bent Writing Institute, Capitol Hill
SPLAB, a spoken word performance and outreach center in Auburn
Fremont Abbey Arts Center, featuring young artists monthly, located in Fremont
Seattle Arts Commission
UW Creative Writing Prizes and Awards
UW Libraries: Information on Publishing Your Work
Association of Writers and Writing Programs
The Poetry Foundation
Newpages.com list of literary magazines
Resources from Poets & Writers:
Advice on submitting your work to literary magazines
Advice on publishing your book
Database of literary magazines
Listing of writing contests, grants, and awards
List of small presses (book publishing)
List of literary agents
There are also some books that provide advice and list venues for publishing your work, including Writer's Market, Poet's Market, Novel & Short Story Writer's Market; The International Directory of Little Magazines and Small Presses; CLMP's Literary Press and Magazine Directory; and Literary Market Place, all of which should be available through UW Libraries.