Hi,
I'm new to this entire thing aswell. I just saw your post and wanted to put in my two cents. I do agree with everyone that the therapy route is good way to go. It will help you get over some of your social anxiety issues.
Some things you can try as a social experiment though is to go out to a bar, party, wherever and talk to just guys to start. When you have no trouble with that talk to people in groups (mix of girls and guys). Then transition into just talking to girls, practice on girls that you aren't attracted to. There is less pressure of rejection since you aren't actually looking for anything. It sounds like a long process, but practice makes perfect. Once you are comfortable talking to girls that you aren't interested in, then you can hopefully be able to move onto talking to girls you are attracted to.
Just remember, you will never see them again if they turn you down (considering you are not hitting on your next door neighbour).
Also, beware of the online dating sites, I went on them, I found 1 descent guy out of like 50. and being bi-polar/borderline makes it harder to have good judgment when meeting people. I met a lot of guys that were just interested in sex, or weren't interesting at all. If you decide to go on the sites, even as another social experiment, just be careful of who you are meeting.
Good luck