Where should I submit a short story if I actually want to see it published?

This is a for a friend, but I’ll ask in first person, because it might apply to me later. If I have a piece of short fiction (or nonfiction) that seems good to me—as in, I’d be happy if I read it in the Sun, or something comparable—how would I go about getting it published? Should I just submit it somewhere (i.e. The Sun)? Or should I choose a journal or magazine more carefully? And how would I submit to maximize my chances of actually getting published?

Any insight from people who have actually gotten work published in reputable places would be appreciated.