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. 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. These include a bit more on real forms, and a bit less on the non-semisimple groups.