Where can I learn about differential graded algebras? I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and compute examples of cotangent complexes of morphisms of schemes.
Where can I learn about differential graded algebras which will provide the computational tools to tackle these questions?
 A: In the same vein as zyx's answer, if you want to learn about cdgas and their applications to rational homotopy theory (before trying to use them in derived AG, which would make sense IMO), I can recommend these two books:


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*Yves Félix, Stephen Halperin, and Jean-Claude Thomas. Rational homotopy theory. Graduate Texts in Mathematics 205. New York: Springer-Verlag, 2001, pp. xxxiv+535. ISBN: 0-387-95068-0. DOI: 10.1007/978-1-4613-0105-9. MR1802847.

*Yves Félix, John Oprea, and Daniel Tanré. Algebraic models in geometry. Oxford Graduate Texts in Mathematics 17. Oxford: Oxford University Press, 2008, pp. xxii+460. ISBN: 978-0-19-920651-3. MR2403898.


There is also this introduction that you can find online:


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*Kathryn Hess. “Rational homotopy theory: a brief introduction”. In: Interactions between homotopy theory and algebra. Contemp. Math. 436. Providence, RI: Amer. Math. Soc., 2007, pp. 175–202. DOI: 10.1090/conm/436/08409. MR2355774.


You also have the original papers:


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*Dennis Sullivan. “Infinitesimal computations in topology”. In: Inst. Hautes Études Sci. Publ. Math. 47.47 (1977), 269–331 (1978). ISSN: 0073-8301. Numdam: PMIHES_1977__47__269_0. MR0646078.

*Daniel Quillen. “Rational homotopy theory”. In: Ann. of Math. (2) 90 (1969), pp. 205–295. ISSN: 0003-486X. JSTOR: 1970725. MR0258031.


But note that the two books I mentioned at the beginning have the advantage of having been written 20+ years after these things were discovered, the subject matter was a bit more "settled" and the exposition is a bit clearer.
The book by Félix/Halperin/Thomas also has a sequel, Rational homotopy theory. II (same authors, published last year) that goes over the same material as the first book but more quickly, and then explain how to apply the same techniques to non-simply connected spaces. The original paper by Sullivan already deals with non-simply connected spaces, he does the whole theory at once; I find it's a bit simpler to learn first about rational homotopy theory of simply connected spaces, then learn about non-simply connected ones.
A: Here is a more recent and concise version on this:
Jenny August, Matt Booth, Juliet Cooke, Tim Weelinck, "Differential Graded Algebras and Applications," Jan 2016, available at http://www.maths.ed.ac.uk/~mbooth/files/hodgeproject.pdf
This also has 31 references, that can help go deep in the sub-part where you need more details.
A: I think that the Manetti's Lectures on deformations of complex manifolds is sufficiently complete about differential graded algebras theory.
A: DGA is usually understood first from the point of view of rational homotopy theory.  The application to deformation problems and the interpretation of that as derived geometry came later. 
Some references from the "classical period":
Griffiths and Morgan notes Rational Homotopy and Differential Forms 
the 4-author paper on formality of Kaehler manifolds (Deligne-Griffiths-Morgan-Sullivan)
and the original papers by Sullivan and Quillen. 
