There are many resources that list places to publish your new work.
The online and print journal Poets and Writers offers reliable, ongoing information about submission opportunities in all genres. Opportunities include literary journals, magazines, manuscript publishing, and grant opportunities.
Another reliable list of publishing opportunities is offered on Entropy Magazine. Here you will find information about upcoming deadlines and links to sources. The list is updated every three months.
Finally, by opening a free account on Submittable, you can search for upcoming deadlines in your field.
Remember, submitting work to literary journals often requires paying a reading fee. Magazines and journals typically ask for $3-$5. Contest and manuscript reading fees are higher: sometimes $20-$50. You do not get these fees back if your work is not used. Read the guidelines well before you hit submit!