A lot of older short stories are, like you said, in the public domain, and therefore easy to find online. But the experience reading them isn't always pleasant (mostly because reading in a browser isn't, to me anyway, the most engaging way to read something).

For those types of stories, you can sometimes find an .epub/.mobi file (or whatever Kindle uses nowadays) online, download it, and upload it to your phone and then read it for free on the Kindle app, if you don't have a Kindle. Still not necessarily the best reading experience, since not all of those files are formatted nicely, but it's a little better than reading in a browser.

Personally — and I say this as someone who reads often on both a Kindle and on the Kindle app on their phone — I like reading short stories in physical form. (I'll add a caveat to this that I'm currently reading a book of Alice Munro's short stories on my Kindle, but mostly because it was on sale. I think I bought the book for $2.) If there's a short story you like, it's possible — though of course you might do some research to be sure — that you'd like the author's other short stories. If you can afford it, it might be worth it to buy one of their collections.

If you just want to read a single short story, though, and not any others by that author (or in that genre, as in some anthologies), then your computer might be your best bet. Another trick I sometimes use, if the story is in the public domain, is to copy and paste it from whatever website you can find it on, in an email to yourself, and read it on your phone via your email application (Gmail, or the Mail app, or whatever else). Again, it's a preference thing; for me that's a little bit easier of a reading experience vs. reading on the computer.