Learning about Lie groups

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). However, if you want to learn about the "real" side of Lie groups, both in linear and abstract manifold setting, my favorite is Godement's "Introduction à la théorie des groupes de Lie".

Several of the books mentioned in other answers are devoted mostly or entirely to Lie algebras and their representations, rather than Lie groups. Here are more comments on the Lie group books that I am familiar with. If you aren't put off by a bit archaic notation and language, vol 1 of Chevalley's Lie groups is still good. I've taught a course using the 1st edition of Rossmann's book, and while I like his explicit approach, it was a real nightmare to use due to an unconscionable number of errors. In stark contrast with Complex semisimple Lie algebras by Serre, his Lie groups, just like Bourbaki's, is ultra dry. Knapp's Lie groups: beyond the introduction contains a wealth of material about semisimple groups, but it's definitely not a first course ("The main prerequisite is some degree of familiarity with elementary Lie theory", xvii), and unlike Procesi or Chevalley, the writing style is not crisp. An earlier and more focused book with similar goals is Goto and Grosshans, Semisimple Lie algebras (don't be fooled by the title, there are groups in there!).

There's also Fulton & Harris "Representation Theory" (a Springer GTM), which largely focusses on the representation theory of Lie algebras. Everything is developed via examples, so it works carefully through $sl_2$, $sl_3$ and $sl_4$ before tackling $sl_n$. By the time you get to the end, you've covered a lot, but might want to look elsewhere to see the "uniform statements". An excellent book.

Brian Hall's "Lie Groups, Lie Algebras and Representations: An Elementary Introduction" specializes to matrix Lie groups, so it makes for an accessible introduction. Like Fulton & Harris, it's got plenty of worked examples. It also has some stuff about Verma modules that's not in Fulton & Harris. I think it'd be a great book for a first course.

Knapp's "Lie Groups: Beyond an Introduction" might be good for a second course (it has more of the "uniform statements" Scott mentioned) and is handy to have around as a reference. It has an appendix with historical notes and a ton of suggestions for further reading. It also has a lot more on Lie groups themselves than most books do.