It’s been a long time since I read the stories: I did thumb through a couple before writing this, but I don’t pretend complete expertise.
On Yiddish (the common language): He would certainly speak Yiddish; that would be the language of the Jews speaking together. Whether he could read or write Yiddish, that would depend on what kind of education his parents could have afforded. IIRC, some of the stories are set as letters from Tevye to Mr Sholem Aleichem (in Yiddish) which implies that he could read and write in Yiddish.
On Hebrew (the language of the Holy Books): Studying would have been oral, textbooks were costly, so he might or might not read Hebrew. He would certainly not know modern conversational Hebrew – that’s a later revision of the language. He would have been taught prayers and bible quotes in Hebrew; in my quick thumb-through, he quotes (or misquotes) Rashi, a biblical commentator. One of the running jokes, in story, play, and movie, is how he mangles them: he’s sort of a walking example of “a little learning.” His longing for studying with the learned men every day is his dream, and doesn’t mean that he had studyed much in the past. The goal of learning was very important, but the families financial situation would presumably have restricted him. At the other extreme, the character Lazar-Wolf in the story clearly hasn’t studied anywhere near as much as Tevye.
ASIDE: I only have English translations of the stories that are in the form of letters from Tevye to Sholem Aleichem, when he mangles biblical texts, I don’t know if he’s quoting in Hebrew or transliterating into Yiddish. If he’s quoting in Hebrew, then he could obviously read and write it.
On Russian: This would be the language for dealing with the non-Jews. Since Tevye (and the one daughter) obviously do deal with non-Jews, they do speak Russian. I don’t think that the example of Isaac Asimov (quoted from Wikipedia by Wendell) is pertinent: he left Russia at age 3, so not knowing Russian isn’t indicative of much. The evidence from the stories, play, and movie is that he did speak Russian. Whether he could read and write Russian, however, is a different issue: the movie chose to make a point of saying no, he can’t read or write Russian. The play (and I think the stories) don’t address the question.)
My feeling: The character Tevye of the stories is reasonably well-educated, even if he misremembers (or misuses) bits. He is also a great story-teller, as the stories are told in first-person narrative. I think it most likely that he could speak at least Hebrew, Russian, and Yiddish; that he coudl read and write Hebrew and Yiddish, but unresolved whether he could read or write Russian.