What if there was a girl who you saw on average 3 times a week, you didn't know her, but saw her in the hall at school, etc.
The one girl, I am thinking of just going for it, and walking up to her and being like, "Hey, I'm [Running Free]. Are you doing anything this weekend? Would you want to perhaps go to [some place for a date]?

Right now, I am thinking that this is the best chance I have.
Comments, suggestions?
Do you think it will work?

Just start talking to her a few times first, and see if you guys have any chemistry. If so, then tell her your going to (event), and that she should come with you.

^^^ What he said. I've found building some rapport is better than just randomly approaching someone for a date.

Yeah, the reason I am thinking of doing this is that the initial approach is what gets me. I have no idea how I should just randomly start talking to her, because when I do see her, it's while both of us are moving, often in opposite directions.

Well, you can either wait for an opportunity to present itself, or...

you can make an opportunity!

I think the first thing you should do is find out whether or not she has a boyfriend.

Right, so wouldn't I find out this stuff about her while out? Not a date, just a time to hang out or something, like going out with friends or something.

i would walk up to her and be like, " hi whats your name?' after that tell her your goin to see a new movie and see if she is free that friday.

Agreed. Throw water or marbles on the floor in front of her. Help her pick up her shit when she falls hard.