Learning Hodge theory is certainly a bit of hurdle. Most people coming from the algebraic geometry side are not so comfortable with the analysis, and analysts may not be so happy with the spectral sequences....
Although I'm in the first category, I've forced myself to go through some of the details as a student. Here's a very rough idea. The basic problem is to show that the space $closed(X)$ of closed $C^\infty$ forms on a compact manifold $X$ is a direct sum of the space of exact forms $exact(X)$ and the space of harmonic forms $harm(X)$. These spaces are orthogonal (easy), so one might try to first prove that
$$closed(X)= exact(X)\oplus exact(X)^\perp$$
and then identify the latter with $harm(X)$. Since these are infinite dimensional, the decomposition isn't automatic. However, one can make it work by using $L^2$ forms and applying Hilbert space methods. But at end one wants to come back to $C^\infty$ forms, and here is where the magic of elliptic operators comes in. The basic result which makes this work is the regularity theorem: a weak solution of elliptic equation, e.g Laplace's equation, is in fact a true $C^\infty$ solution. As you said, the full details entail quite a bit of work, but one can look at standard books on Riemann surfaces (Farkas-Kra, Forster, Narasimhan...) for some instructive special cases.
One last comment, the method of Deligne-Illusie is wonderful, but as Ravi Vakil commented, it is not that easy. Nor does it yield the full Hodge decomposition. Incidentally, the first algebraic proof of degeneration of Hodge to De Rham was due to Faltings. Deligne-Illusie appears easy only in comparison.