I wouldn't consider a movie date with a stranger that far into the distance to be an actual *date*, but rather very tentative plans. If you want to see if this girl likes you, you will have to get a REAL date sooner.
As for your mom, you are 19! She won't be surprised that you are interested in dating. Just tell her that you have a date with a girl you met. I doubt your mom is worried about you being raped, unless you are female.
Relax... I'll need some information first. Just the basic facts - can you show me where it hurts?