Argyres' 1996 supersymmetry lectures, through lecture 7, provide a good introduction to the ideas
and the basic technology.
They can be found here.
(There's also a 2001 edition, but I haven't read those.)
The rest of the lectures are also very valuable, but not as immediately necessary for the string theory class.
Also useful, but much more austere, is Wess and Bagger,
through chapter 8 (and not any further!!).
Appendix B of Polchinski is also useful but terse.
Chapter 4 of GSW has a discussion specific to our worldsheet needs, and not coincidentally uses the same notation as on pset 4.
The structure of 2d (2,2) supersymmetry (and its relation to 4d N=1) is explained clearly in The Phases Paper. Its conventions are used in the penultimate problem on pset 4.
A free, long, encylcopedic book is Gates Grisaru Rocek Siegel, Superspace, or 1001 Lessons in Supersymmetry.