What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?

Of course, since I haven't found a (for me) readable introduction, I don't know what I should learn about Hodge theory, but at least I'm looking for:

reasons to expect that solutions to the laplacian (rather than other operators) should be studied

examples where the harmonic forms can be explicitly written down

how to think of harmonic forms in general

the hodge diamonds (and how to compute them!) of the standard examples of projective varieties

So far I have read (parts of) the relevant parts of Griffiths-Harris and lecture notes on the web, but still don't how understand how to do computations with Hodge theory, or for which varieties I should expect to be able to use it.