READING ACTS: Using Rabbinic Literature for the Study of the Second Temple Period. The best advice is probably not to use rabbinic literature for the study of either the New Testament or the Second Temple Period unless you are a specialist in Rabbinics. But people generally are not going to take that advice.
Two comments. First do not use Strack and Billerbeck. See Phil's note in the comments for why. Second, the only way I know of to draw on the Mishnah on its own terms for information on the late Second Temple Period is to restrict the discussion to the sayings that Neusner and his students reconstructed as belonging to the earliest stratum ("before the wars"). This reconstruction is itself open to debate in various ways, but unless you want to specialize in Rabbinics and form your own opinion about the date of each saying in the Mishnah, it's the best cheat sheet we have at present. (This is outside my area of expertise, so if you know of a better one, please alert me to it.)
Relevant basic bibliography:
Neusner, Jacob. Judaism: The Evidence of the Mishnah. Chicago: University of Chicago Press, 1981.
Sanders, E. P. Jewish Law from Jesus to the Mishnah. SCM, 1990.
Yes, I know that Neusner and Sanders disagreed on a lot of things, but Sanders's overview of Neusner's earliest Mishnah stratum in the book above is helpful.
Past posts in Phil Long's series on the Second Temple Period are noted here and links. I needed to look some things up before I published this post, so I have taken the Reading Acts post out of order.
Finally, for more on teaching the Mishnah to undergraduates, see here.
UPDATE (6 February): Richard Bauckham has e-mailed to remind me of David Instone-Brewer's still-in-progress work on that first stratum of rabbinic sayings: Traditions of the Rabbis in the Era of the New Testament (T-R-E-N-T). And see also here.