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.