I don't know if it's appropriate to link to self-advertise here.  So at the risk of a minor faux pas, my edited notes from the Lie Groups class taught by Prof. Mark Haiman are available [here](http://categorified.net/LieGroupsBook.pdf).  The bulk of the notes is the classification of complex semisimple Lie groups.  For compact ones, follow the same argument, but add one fact: a simple group over R is compact iff the Killing form is negative definite.

In case one should not post one's own notes, [here are some by Anton from the previous year](http://math.berkeley.edu/~anton/written/LieGroups/LieGroups.pdf) ([Wayback Machine](https://web.archive.org/web/20130521124328/http://math.berkeley.edu/~anton/written/LieGroups/LieGroups.pdf)).  These include a bit more on real forms, and a bit less on the non-semisimple groups.