I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ or $Y(2)$ or whatever doesn't matter) from the Betti point of view ($SL_2(Z)$, representations of the fundamental group, etc) and the de Rham point of view (Picard-Fuchs equation, hypergeometric functions, etc). This is for a student who has taken classes in algebraic topology and complex analysis and who is just learning algebraic geometry, so I would prefer something that's as down to earth as possible, not a full-blown sheaf-theoretic treatment (i.e. with $R^1f_*$, D-modules, etc). This is a beautiful classical subject, so I can't believe there aren't really great expositions out there, but I can't think of even one!
[Edit 2010/01/21: Thanks to every who suggested references below. I'll probably suggest Clemens's book and Hain's notes. But after looking at them, I realize that I'd really like something even more basic, with no algebraic geometry (cubic curves) and no holomorphic geometry (Riemann surfaces). I just want the moduli spaces of homothety classes of lattices in C (maybe plus some level structure), viewed first as a topological space and later as a differentiable manifold, together with the H^1 local systems, viewed first as a representation of the fundamental group and later as a vector bundle with connection. Probably this is so easy that no one ever bothered to write it down, but on the chance that that's not the case, consider this a renewed request in more precise form.]