In my online journeys in search of reputable publications here are just a few that I have gleaned from the cyber fields. Most of them have been around for quite a while. That fact alone means they must be doing something right.
Ploughshares A publication of Emerson College. View their submission guidelines. Share information about their Emerging Writer's Contest with an aspiring writer.
All Aboard for the Glimmer Train! This is a creative literary journal with a longstanding readership. Polish up that short story you've been meaning to send in and submit it. You can view their guidelines for submissions here and find out more about their Family Matters contest, and how they help new writers get off to a good start.
The Missouri Review is another publication where new and previously published story writers may get heard. Info about submitting to regular issues and contests can be found here. All those poets and writers out there should give it careful consideration.
For a more comprehensive list of publications you may want to check, and even subscribe, to the publication titled Poets & Writers and check out their online database of publications. This database lets you search by genre and subgenre so you can more effectively focus on publication best suited to your particular writing or poetry. There are so many to choose from.
Hope these little tidbits help. Have a terrific day.