I am currently trying to learn this, and [this paper][1] of B. Keller proved very useful. [This one][2] of J. Huebschmann seems a bit less basic, but might be usefull too. I must say I'm very interested in any other answers.

  [1]: https://arxiv.org/abs/math/9910179 "Bernhard Keller: Introduction to A-infinity algebras and modules"
  [2]: https://arxiv.org/abs/0809.4791 "Johannes Huebschmann: On the construction of A-infinity structures"