Best Versions of Chekhov?

NONE OF THEM. But if you have to read/watch a Chekhov play, pray to the baby Jesus that it isn't the Mamet adaptation of Three Sisters. Absolute torture.

This is probably not the most helpful answer, but the thing that finally made me "get" Chekhov was reading his work in the original Russian.

Assuming learning a new language is out of the question, I am partial to Curt Columbus's translations. His translations capture the bluntness of the characters and the directness of the language. The most important thing is to avoid the "translations" by non-Russian-speaking playwrights who freely adapt literal translations. I've never seen one that really accurately reflected the feeling of Chekhov's language. On the other end of the spectrum, you also have a lot of highly accurate translations by academics, which often don't scan all that well in English. I like Columbus's translations because I think he balances the two extremes very well as both a fluent Russian speaker and a talented playwright himself.

But reading Chekhov is never going to really get at what's so great about his work--you have to see it performed. Unfortunately, I find it so rare to find really well-performed Chekhov. So many productions entirely miss that the man wrote comedies (although with some of the translations, it's understandable how you'd miss that). I really enjoyed the film of The Seagull that came out last year because of how much of the humor came through. It's not perfect, but it made my mother, a novice to Chekhov, say "I thought Chekhov wrote boring, sad plays where nothing happens--this movie was hilarious!" and I'd say that's a win. Watch that, and watch Vanya on 42nd Street, and if neither of those really clicks with you, then Chekhov probably just isn't for you, which is okay.