The following is an emended excerpt from my answer to a related question^{1} about books about Lie groups for someone with algebraic geometry background. I might add that Procesi's book ideally fits your goals, since you are also interested in representation theory.

For someone with algebraic geometry background, I would heartily recommend Procesi's *Lie groups: An approach through invariants and representations.* It is masterfully written, with a lot of explicit results, and covers a lot more ground than Fulton and Harris. If you like "theory through exercises" approach then Vinberg and Onishchik, *Lie groups and algebraic groups* is very good (the Russian title included the word "seminar" that disappeared in translation).

If you aren't put off by a bit archaic notation and language, vol 2 of Chevalley's *Lie groups* is still good.

^{1}That question is exactly one year old and, according to Anton's MO birthday post on meta, was the second "real" question asked on Mathoverflow.