I agree with Annie that the best way to meet new people is to join a group related to a hobby or something else that you really enjoy doing. Since all members of the group will enjoy doing the same activity, it provides the chance to not just meet new people on a superficial level but also to develop some deep genuine friendships, or maybe even something more if you're single and looking for love.
Regarding making friends on Facebook...I'd personally be very wary about that. Without actually meeting people face-to-face, it would be very difficult to know whether the person you're communicating with is who they say they are...and this has had disastrous consequences for some people.