I've been a big fan of Wittgenstein since college, and have at this point read just about all of his work. I might suggest that people new to Wittgenstein start neither with the Tractatus nor with Philosophical Investigations, but with the Blue and Brown Books (you want the Blue part).

These are lecture notes toward what would later become the PI. They're way less systematic than the big two, but they'll give you a great feel for Wittgenstein's thought.

I'll also second Monk's biography, which is just superb.

One of my favorite things about Wittgenstein, is that there's a powerful strain in his thought that regards philosophical speculation as essentially a form of "madness." That is to say, if you're sitting around wondering what "justice" is or trying to figure out the mind/body distinction, you're engaging in a kind of nonsensical behavior. It is the purpose of philosophy to resolve these problems -- which, in Wittgenstein's case, often meant showing why they're nonsensical questions in the first place -- so you can go back to living your life. In Monk's biography, you often see him revolutionizing modern philosophy and then afterward going to work as a gardener.

I think this explains, in part, why he is more admired than influential in modern analytical philosophy. He is one of the many twentieth-century philosophers to suggest that the proper outcome for philosophy is to stop doing it.

[edits for typos]