Thanks for your help, guys. Just one more question: should I ask her if she wants to go somewhere casually, or should I just go straight to immediately asking her out? I have people saying both ways, and am not sure which is the better approach.
I'm not too sure what you mean.
You can keep it casual but put your true intentions out there. Let her know that you're interested in spending time with her. I mean, you two have known each-other for a good time, so I don't think that you'd have any problems just asking her word for word "Would you like to spend some time together?"
There is a gray area that I'm working around. Have you expressed any interests beyond just being friends, or is does she not have a clue? If she has an idea that you like her, word it like I've said, but if not, I'd just ask her if she's interested in doing something (Food-Hanging yada, yada).
You can get her out and about with you, but letting her know that you're interested her while with her could go from 0 to awkward very, very fast. Best bet is to just ask her if she'd interested in a relationship. I wouldn't fear the initial rejection, if I were you. You have to work with her, and the fact that you've mentioned that she's reserved and shy may make it hard for her to communicate her feelings if you do get to rejected.
After saying all of that. Like Slip has said, it's best to wait until the dust settles after the group meeting, but make your intentions kind of clear before everyone starts leaving. Just before everything is about to wrap up, ask her if she's busy and if she's interested in hanging out or whatever you want to do.
If she says she's busy or wanting to be alone, accept that, but ask her if she'd be up to doing something later on. That'll make it pretty clear that you're interested in her because of you being somewhat head-strong.
Good luck, dude.