In the simplest form, look for dependant clauses. That is, can it stand alone as a complete sentence or is it dependant upon the rest of the sentence to make sense.
Now, obviously, there are exceptions. Two of which you see here. "
That is", technically, is a complete sentence. It has a subject and a verb. The proviso here is that we really don't know what "that" is unless we take in the rest of the sentence. The second instance is the interjection. Not like an "OH!" or "Alas" that you might see or hear. This is more of a construction matter. "
Obviously" is not part of the body of the sentence and would not make any sense just hanging out there by itself. It is solely dependant upon some other comment to support it.
"That VW is not big enough for all six of us," he says as the group tries to squeeze into the little Jetta.
She favors him with a withering glance and says, "Obviously."
Obviously what? Obviously, the Jetta is not big enough to hold six people. Her comment is a complete (though contracted) statement but we would have no idea what she was talking about without his previous comment.
Now, as to that 'introductory comma' you wondered about.
When you find a dependant clause at the opening of a sentence or, if you find the beginning and the end go together but there is a dependant clause dividing the main sentence, that is where you are going to find your commas.
Now, another conundrum you may encounter is the compound sentence. Something like, "Instead of brussels sprouts, which Kaylie declared she hated, Mom cooked broccoli." [Note that the inclusion of the word "which" makes the dependant clause dependant upon the main sentence, "Mom cooked broccoli." Ahead of all of that, however, is the clarifier, "Instead of brussels sprouts," (also a dependant clause).
Totally confused yet? I'll try to simplify. If it can't stand alone, it's going to need a comma.