"They study the first unit."
Translation:Sie lernen die erste Lektion.
It depends a bit on the notion. Studieren is used for university studies and for carefully reading [e.g. the news] or looking at/examining objects. It's not a very clear case here to exclude studieren, but I would definitely prefer lernen in the context of unit [of a textbook = Lektion]. For another kind of unit it could be studieren as well. If you can say "The generals are studying the first unit of the coorps passing by at the parade", you can do the same in German with studieren... A less likely notion, but not impossible...
The question was: "They study the first unit." A couple of correct translations are:
Sie lernen die erste Lektion. / Sie studieren die erste Einheit.
Teil (usually translated as "part") seems a little too vague and possibly too large, as it is likely to contain several units.