On the OP:
Basically, the advice is to pay attention to the holdings of the cases. (The rule in the case is called the "holding" in technical legalese.) This is a reasonable first-approximation approach, but it can be tricky depending on how the professor sets up the class. A lot of cases are not just illustrations of a well-established rule; they're on the margins of an established rule, and especially in the core common law classes (Contracts, Property, Torts), there are state-by-state differences. A case may be assigned so that you can see one way a rule can play out, and then the professor may tell you (or may assign a second case that illustrates) the other way that rule could play out. In your assigned cases, you might have multiple contradictory rules, or an awkward adaptation of a rule, or any of a number of other things, and the casebook may or may not ever tell you whether the approach given in the case is particularly idiosyncratic.
At some point deep into personal jurisdiction in Civil Procedure (Pennoyer -> International Shoe -> etc.), I finally decided that you just can't tell what proposition a case is going to stand for until you hear the prof talk about it and see how later cases cite it, so I gave up writing down holdings in preparation for class. My briefs consisted of header (name of the case, court, year, page in book), facts, procedure, stuff the court said about the law and how it applied, and the result (the disposition of the case). Instead of writing down the holding right away, I starting writing down the rules that cases stood for about a week or two after class, when it was clear in context what the heck we were doing that day. I kept this up through all of 1L, and I'm not sure if I'm going to change it for 2L. At least for the first semester, nothing made sense to me at first, but it all clicked when I came back to it after having seen the surrounding material.
Also, my Leg Reg class didn't depend on holdings really at all, though admittedly 1) Leg Reg isn't a standard 1L class and 2) I think it was somewhat unusual even as a Legislation/Admin Law class. So this can vary a bit from professor to professor.
I think some time in August I'll type up a big fat post with everything that I wish someone had told me as I was entering first year. Stay tuned.