It's really hard to narrow these down. They are all so unusual that it's hard to pick favorites. But, here are the ones that are most talked about, along with a couple that I really liked but aren't frequently mentioned.
A Trifle from Life
Hush!
Champagne
The Kiss
The Grasshopper
Ward No. 6
Gooseberries
The Darling
The Lady with the Dog
After having written that list, I really do recommend reading all of them. The whole book is only 337 pages, and every story seemed to give me some new insight. Ward No. 6 was the only story on my list that was a little hard for me to get through, but it was worth it. I don't know if you need to read the introduction first, it's probably better to read it after the stories. But I liked these quotes from the introduction:
"If anything can be termed 'typical (Chekhov)' it is that he insists we keep our notice close to life's nuance, it's intimate gestures and small moral annotations."
"As readers of imaginative literature, we are always seeking clues, warnings: where in life to search more assiduously; what not to overlook; what's the origin of this sort of human calamity, that sort of joy and pleasure; how can we live nearer to the latter, further off from the former? And to such seekers as we are, Chekhov is guide, perhaps
the guide."
I don't know that I'd call him
the guide, but he is pretty remarkable. Read away and enjoy!