What are some good lit magazines to start publishing short stories?

What type of writing are you primarily interested in publishing? This plays a significant role in where you should look. Some magazines put out calls for only non-fiction, some want short stories of a certain length, some specifically cater to LGBTQ audiences, some want stories about nature. What you want to write is something you should keep in mind while looking for magazines.

Poets&Writers keeps an updated database on magazines seeking material. This is one of the places you should start sifting through. Keep a journal open on your desk and write down the names of the magazines that catch your interest. Pay attention to what type of journal it is, the genre of material they want, how they accept their submissions, and the deadlines.

Don’t be intimidated by big-name journals. It doesn’t matter whether you send your work to a never-heard-of online journal or a New Yorker-type. My poems were rejected from a big-name journal and I’m perfectly fine :) No ego-bruising whatsoever. You dust yourself off and send the work to a different journal.

Another thing you need to jump on is writing cover letters. I’m terrible at writing cover letters still.

You may want to look into getting a copy of the latest Writer’s Market. (A local library might have a copy you can look at for free.) It’s pretty much the standard guide to what magazines are looking for, what they might pay, how to submit work correctly, etc., and it’s updated annually.

The Kenyon Review is one of the most highly regarded literary journals.

In addition to the above suggestions, I would say (a) consider your own school’s literary magazine, if any, (b) look for opportunities to publish in less prestigious outlets such as community newspapers and online sites for the sake of the experience, the publication credits, and the exposure,  (c) ask your instructor to recommend publications with which you might have a good chance of acceptance, and (d) submit entries to contests (especially small ones), knowing that your entries will be read and that you don’t have to have a pedigree to win recognition that way.

Reader’s Digest. Sounds odd to add that but if they publish it they send you money.
Depends on what you are writing.

I agree with Frenchfry, go to Reader’s Digest

I would think you’d have a better chance of publication in The New Yorker than in Reader’s Digest. Most publications other than small literary and university pubs pay for stories, but they don’t have the global base of Reader’s Digest, Moreover, anything suited to a literary journal is not going to appeal to RD’s audience.