you could have told her, "my mom won't drive me anywhere" instead of "maybe another day". thats what forgeted you over.
Yeah, should've been honest and said you're having problems with transport.
At the very least she might have suggested you do something a bit closer to where you are.
OR, she might have asked one of her parents to pick you up with her and drop you off at the cinema.
There's no point being embarassed about the fact that you can't get to where you want to be. And it's better to be truthful in that than say that you can't make it. From that point on it seems like you're putting other things ahead of the chance to hang out with her, and if it keeps happening it seems like you don't care.
The best thing you could've done was told her, and suggested that you perhaps do things together after school.
I assume you both get picked up by parents, or live in walking distance of your school.
All you'd have had to do was decide to go and do something with her after school, then you both could've walked there, or been driven there by one of your parents. And if all the parents know you're somewhere, they're going to be happy to get you back home afterwards.
That way you get to spend time together, you meet her family, both families know where you are, you get a free ride home and you're both safe. It'd be a complete win-win situation for everyone.