Yes, we hire new grads, and we offer internships for those who are still studying. Successful interns get an easier path for converting to full-time than grads who apply cold.
Getting somebody you know at Google to refer you is the best way to secure an interview, but you can also apply online.
The hiring process is a bit hit and miss. I know people who had a really good experience, and I know people who said it was a shit show. About meeting Googlers: if you happen to come to the Bay Area, you can't throw a rock in any direction without hitting at least one Googler. Not saying throwing rocks is the way to go, but... you'll get someone's attention :P
I've noticed that Google is better than most with online applications. I managed to interview 2 years in a row (intern, college student) having only applied online.
