Here's a slightly expanded version of my previous answer:

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 when I was 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 $harm(X)$.  A closed form is harmonic if and only if it is orthogonal to closed form (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 to obtain:

$$\overline{closed(X)}= \overline{exact(X)}\oplus \overline{exact(X)}^\perp$$

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.
The space on the right of the second decomposition is therefore $harm(X)$, and this
is all one needs.

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. Also at the risk of some shameless self promotion, I may as well admit
I  have some notes of my own

http://www.math.purdue.edu/~dvb/preprints/book.pdf

with more details (although I use the heat equation rather than orthogonal projections).



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.