I think most social people really don't mind talking to strangers. In fact if you are non threatening and get them to talk about themselves they really like it. People like to talk about themselves, it's almost instinct.
It's really not that difficult, just takes a bit of practice. The hard part is finding the right tone of voice for maximum effectiveness.
I don't think seeming like you care is a big no-no though, caring about what another person is saying is very important. Being interested in them, by asking questions about what they are into and how they feel and such and such.
Also going from casual to having enough nerve to ask for a number or a date or whatever... to continue elsewhere. That's weird for me, I don't know really where to drop it in and not have it be awkward. It's difficult for me to ask anyone for their number though, men, women... anyone.
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Wonder what it feels like to be in love?
How would you describe it, like a push or shove?
Guess I could pretend that this is all I need
Wanting more than what I have might appear as greed.