Griffiths and Sullivan wrote a fine book on the subject. Apart from the obvious attractiveness of learning a theory from its creator, it is written in an amazingly user-friendly style. For example, Chapter XIII is devoted to examples and computations: it starts with the computation of a minimal model for the forms on a sphere
and ends with Massey triple products on compact Kähler manifolds, a section inspired by the 1974 Inventiones article of Deligne,Griffiths, Morgan, Sullivan. The first hundred pages (Chapters I to VII) are an introduction to the necessary algebraic topology and you can probably essentially skip it, judging from your description of what you already know.
Reference Griffiths, P.; Morgan, J. (1981), Rational homotopy theory and differential forms, Progress in Mathematics, 16, Birkhäuser