I think I want to spend some time reading some good short stories. Unfortunately, I don't have the money to buy any. So where can I go online to read a variety of top-quality short stories in various genres?
Blackbird is a really good online publication to read short stories. A lot of publications will have some free content on their website.
There are all kinds of short stories available via a quick google search. You just have to know what author you're looking for. Generally, it's older and more famous stuff that you'll find as PDFs. And by older, I don't mean archaic. I'm sure you'd be able to easily find Flannery O'Connor, Raymond Carver, Ernest Hemmingway, etc. The same authors you'll likely find in literature textbooks.
What genres do you like? As @deadrats mentions, a lot of magazines offer at least some content for free. I'm mostly into sci-fi so I'd recommend Lightspeed, Clarkesworld, Newmyths (totally not just 'cause they published me ...), Uncanny - I should be able to come up with more but it's been a long day. And I won't say they're good, but I read a lot on r/nosleep. If you like horror, you'll find some gems in there now and then.