Surveys of Goodwillie Calculus

It looks like the nLab article is pretty nice. There you'll find a list of references, including this:

Brian Munson, Introduction to the manifold calculus of Goodwillie-Weiss, arXiv:1005.1698

Is that the kind of thing you're looking for?

You can get a pretty good birds-eye view by reading this 2004 Oberwolfach report (no. 17):

http://www.mfo.de/document/0414/OWR_2004_17.pdf

Calculus of Functors is the theme of the 2012 Talbot workshop, and one can hope that there will be reasonable attempts made to collect introductory and expository material and make it available. There may be much better answers to your question in a month or two (written 1/23/2012).

This is 7 years too late, but this survey (to appear in the Handbook of Homotopy Theory) is a really readable survey of Goodwillie calculus: https://arxiv.org/abs/1902.00803.