Social Theme Like Facebook, Twitter

You’re looking for plugins for that, not themes. Themes change how a site looks, plugins provide functionality.

For what you’re describing, I recommend https://buddypress.org/

Is buddypress customizable? I mean I just want to create groups, private messaging for members and members can sign up after they pay for membership?

I recommend asking at https://buddypress.org/support/ so the BuddyPress developers and support community can help you with this.