In the age of the Internet, finding possible publication venues is relatively easy.
Here are several great resources for publication information:
Welcome to Duotrope, an award-winning, free writers' resource listing over 3650 current Fiction and Poetry publications. Use this page to search for markets that may make a fine home for the piece you just polished. Use the menus at the top and right to explore the rest of the free services we offer writers and editors, including a free online submissions tracker for registered users. We usually make several updates per day, and we check each of the current listings regularly (at least once a month) to ensure the most up-to-date database humanly possible.
Duotrope does a great of keeping tabs on old and new markets as well as including specific calls for certain kinds of submissions.
New Pages is a hub for all things literary. Included in that hub are frequently updated calls for submissions and contests. New Pages also has a Young Authors Guide, a resource that's great for student writers.
Poets & Writers-
Poets and Writers magazine has both a print version (paid subscription) and a free online version. Both versions include information about publication, contests/grants and awards.
The online contests/grants and awards section is easy to use. The publication section (under "classifieds") is a bit cumbersome.
There is also a web page of Creative Writing Contests that can be helpful. Additionally check out fiction contests and opportunities.
If you're looking for general artist opportunities, check out Mira's List.