Recently I've coincidentally run into various relatives of the moduli stack $\mathcal{M}_g$ in several unrelated contexts. I tried reading Harris and Morrison's "Moduli of Curves," but it seems to require a substantial amount of foundation just to read the terminology. I can follow the definitions, of course, and I do want to learn all of the foundation in the long run, but I get the impression that, at least in some cases, these things are pretty straightforward, almost classical geometric objects, which leads me to wonder if I could learn about moduli of curves first with lighter machinery.
Is there a quick way to get a feel for these things -- possibly a text that constructs them using less machinery? Or should I just sit down and learn about schemes and stacks and representable functors and such? (I know the definitions of these things, but I imagine it takes some time to really get used to them; for instance, I can follow a proof using Yoneda's Lemma line-by-line, but I don't yet have a good intuition for what it's "really" for).
For the record, I also do not know any Teichmüller theory.