Most Helpful Guy
No... don't facebook... too impersonal.
He kind of made the first move... so if you like him... it your turn to step up!
Next time your are at the gym... ask him afterwards if he would like to grab a drink of some sort to further your first conversation. That way it looks like your just picking up from where you left off.
Most Helpful Girl
Of course it's okay. It's not common in some places, but there's nothing wrong with that.
I think you should ask him if he wants to meet up over the weekend and grab lunch. You can send it over FB or you can say it in person the next time you're in. :)