The First Temple period is still covered by Tanakh. Tanakh finishes off just as they've gotten the Second Temple going.
Yes, there are broad categories. Timewise, we refer to the following periods, very roughly:
Tannaim -- those who wrote the first stage of the Talmud (e.g. the Mishna). This starts in the early second Temple period (though we don't have a lot of writing on it), a few hundred years BCE, and ends around the year 200, when Rabbi Judah the Prince edited the Mishna. That's the core framework of Oral Law. Most questions of how to legally interpret Biblical verses are settled by this point. E.g. by the end of the Tannaim, we know that "an eye for an eye" is just a figurative expression for monetary payment. No one after this point in history will challenge that. (Though they may argue that the verse was also indicating that morally, an eye for an eye is what's deserved, but no one would practically say that's what should be done.)
Amoraim -- the next stage of the Talmud, generally commenting on/interpreting the Mishna. (And deciding which opinion in the Mishna to follow.) That wraps up around the year 500.
Gaonim -- roughly 500 to 1000. (Technically there are the "Savoraim" in the middle there someplace, making some final edits on the Talmud, but that's really nitty-gritty.) They tend to write responsa, in which their answers cite the Talmud. Occasionally we'll see a few Talmud commentaries, but not that much.
Rishonim -- from around the year 1000 until 1500. Three things happen right around 1500: the printing press becomes popular, so we're seeing things printed, rather than handwritten; the Jews are expelled from Spain, which causes sociological shifts; and Rabbi Yosef Karo writes the Shulchan Aruch, the most-popular code of Jewish Law.
Achronim From 1500 until -- well, the disruptions of the 20th Century generally have caused us to demarcate a new era.
Functionally, you'll see the Rishonim writing primary commentaries on the Talmud. E.g. the Talmud says you should rinse meat before salting it to remove the blood; if you look in Rabbi A's Talmud commentary from the 1100s, he'll tell you the reason is "to get the grime off." If you look in Rabbi B's Talmud commentary from the 1200s, he'll tell you the reason is "to soften the meat so soaking will work better."
At the time, most either wrote commentaries tied to the text of the Talmud, or responsa (answers to questions, where the answers will cite the Talmud.)
As a young man Maimonides wrote his own commentary to a few books of the Talmud, but we don't have much of that. What he originated was a free-standing "code of law" (with a different organization structure than the Talmud), where you have to go back and figure out "what piece of the Talmud is he based off here?", and "how did he interpret that piece of Talmud?"
In the 1500s, we saw the Shulchan Aruch saying that generally speaking, we won't propose new legal interpretations on the Talmud beyond what's already been put there; the question is now which of those interpretations over the past 500 years should we follow, and how to interpret those interpretations, as well as new questions that arise. In this example, we find in the Shulchan Aruch: "you should soak the meat for half an hour, to accommodate the medieval opinion that the reason is softening; however, if you were really in a bind, you could follow the opinion that the reason is to remove the surface blood, and therefore a good rinse will suffice."
Since then, we'll still see responsa, commentaries on the Talmud, and commentaries on the newer legal codes, but they are expected to reference the prior periods' works.
