Yup, if you move to a new city after graduating it's pretty hard to get a social life. The best thing I've found is to take up boxing. I did so recently and I'm meeting a lot of people through it. People see me working out and talk to me afterward because they box(ed) too, or we talk about pro boxing, get invited to parties etc. It's only been three months and I'm doing this for the boxing and not to meet people but I'm surprised at how easily meeting people has become (i'm not officially diagnosed but I have self-diagnosed myself as having social anxiety).
So I think the option of taking a class or volunteering at something related to something you're interested in will help. If you aren't interested in the subject or activity you will have a hard time relating to the other people there and it won't work that well. So find a hobby and then find a venue to connect with others who've the same hobby. I will soon also start attending user groups for software stuff so I can network with other software developers, and also make friends that way.
I moved to NY a year after graduating and it was a long time before I made friends. I'm not the type to hang out with coworkers, not that they are bad people, I think they are all cool and some of them would probably be fun to hang out with outside of work, but work life and social life to me must remain separate. I don't want to get drunk and act the fool, and have stories about the stupid things I do be gossiped about at work. Likewise, even just getting to know people on a personal basis introduces the possibility that someone will pretend to be your friend to your face and then get bored and tell people at work random things about you like "man that guy's apartment is messy, he had 5 day old cooked hot dogs on his foreman grill when I went over there once." (that really happened to me). Even though I'm a pretty chill guy myself other people have issues (in general I was surprised to find out the abundance of people with issues in the real world once I moved out for college) and I don't want the possibility of someone else's inter-personal issues affecting my work life.
Making friends with your neighbors -- I dunno about anyone else but I've been in NYC for four years and that's not something that happens unless you're a gangster and they are gangsters too, or you're a young party kid and they are young party kids too. There aren't a lot of people my own age in my hood so my experience may be uncommon. If you don't already have friends in NYC, the abundance of things to do and places to go is useless to you. I don't know anyone who goes places alone and meets people there. I'm talking about things like bars and restaraunts.