I know the fact that stories composed in Greek have Jesus quoting the Septuagint doesn't mean he actually did, but since I've also never seen anyone say Jesus wouldn't have been likely to know Greek, I'm at a loss to guess what version of the Bible would have been most familiar in his social milieu. Partly I'm wondering if popular religious values already included scholarship before the Temple was destroyed (I imagine the traditional answer is no, but I'd like some more context), but I'm also just curious about the state of literacy and multilingual proficiency in the Levant in late antiquity.
I asked this question before without getting a response, but I haven't given up hope. (Shavua tov, everyone!)
You're right, the New Testament is written in Greek, even though Jesus would have spoken mostly Aramaic. This is true of everything he says, not just the biblical quotations, but that's not really any different from, say, the fact that there are plenty of American movies set in other countries where the dialogue is in English-- they're not trying to make a claim that all those people spoke English, they're just telling a story, and you're meant to assume that it's in whatever language makes sense. The gospel writers probably knew the Septuagint best and were writing in Greek, so they used it for the Bible quotes, but they definitely don't hide that Jesus spoke Aramaic, they sometimes quote a word or phrase he said in Aramaic where they feel that the original is important. The Septuagint was definitely around, and commonly read by Jews. Both Philo and Josephus quote it, and Paul, only fifty or so years after Jesus, uses it as his authoritative text of the Bible.
The Targumim (Aramaic translations/paraphrases of the Hebrew Bible) were still mostly oral at this point, so Jesus had probably heard them and probably quoted the Bible in Aramaic, but they weren't published or formalized until later. Onkelos, for example, was born right around when Jesus died.
There was certainly already some culture of learning, although it is hard to know exactly how much Jesus was a part of it. Paul, for example, when he talks about having been a very successful Jewish Pharisee, talks about how learned he was and how he studied with the best teachers, so we can tell that he thought that was an important part of what it meant to be a good traditional Jew-- and that was still before the destruction of the Temple. The gospels were probably all written after the destruction of the Temple, so it's a little hard to know whether this was original to Jesus or based on the Jewish communities 100 years or so later, but they have the disciples refer to Jesus as "rabbi," which shoes that that was a Jewish term of respect by then.