I also moved recently, to a place that I didn't think I had any friends. It turned out, once I started networking through people close to home, there were lots of people that I already sort-of knew here. It was just a matter of having my close friends say, "Hey, my friend Atmosfear just moved to town... you should totally hang out!" and then meeting for drinks or whatever. Also, don't underestimate the power of Facebook; I think people are always happy to see an old, familiar face.

As has been mentioned, the easiest way to find people like you is to join groups of people doing what you like. If you like to play sports, find a team or a pickup game. If you like to volunteer, join a volunteer organization that does something you're interested in. If you're into it, join a local church or youth group ("youth" now extends to your 20s in many church communities.) And if, like Scarf said, you're a geek who doesn't like those things, use Reddit or Meetup or similar sites to find activities specific to your interests.

In my experience, as a guy, replacing guy friends is much more difficult than girls; girls are, by nature, more open to this type of thing than guys. I know that, if I were with a group of friends, and some dude approached me for conversation that I didn't know, I'd assume that he was hitting on me, and look to end it as soon as possible.

Personally, I could never do the whole "go to a bar alone" thing. It's one thing to go to a restaurant and eat at the bar alone (which I don't mind, particularly while reading a book), but I just don't think I would enjoy drinking by myself, trying to find people to talk to.