If you're looking for laugh out loud funny moments or jokes, then this sort of comedy is hard to find (most would argue not there at all) in The Three Sisters. Nor does the play fit the traditional structure of a Comedy, in which events sort themselves out, sometimes ending in the marriage of the play's lovers. Chekhov was a master of character study, of depicting the subtle qualities that make us the quirky human beings that we are. His plays don't really proceed in a conventional beginning, middle and end, they depict, rather, a slice of life.
Some of the humor or comedy of the play comes from the sisters' striving to leave their hum-drum small town life and get to Moscow. They are constantly deciding to leave, but never go anywhere. The comedy comes from listening to the characters, yet again, declare that they're going to Moscow, or hearing Irina (who hates to work), proclaim that work is everything and the solution to life's problems is work. This has a subtle humor, but is also a bit sad, bringing in the play's pathos.
As for tragedy, I don't think, at least in it's dramatic sense, that this word works for any of Chekhov's plays. Disappointing things happen to the characters here, but you get the sense that there life will pretty much go on as before, which negates the earth-shattering alterations and learning of lessons expected in a Tragedy. I would call it pathos and not tragedy. The lack of meaning that the characters find in their lives is an emptiness that the audience can feel along with them, and this emotional connection, audience with character, is called pathos.