I started talking to this girl, Kirah, about a week ago. We've known each other slightly since then, but we never bothered to talk to each other. I'm not sure if she likes me, but it seems like she enjoys talking to me.

When is it best to ask her out? Later, when we know more about each other? Or anytime from here on out, now that we started talking somewhat? Will she still go out with me if she doesn't affirmatively "like" me, but still thinks I'm cool?

when you know your ready for a relationship, and when you know you both like eachother

Im in the same boat. Waiting for this thread to show something.

Wait a while until you get to know each other. There's nothing more awkward then rushing straight into a relationship. This shit is just common knowledge, sheesh.

Depends. Are you in high school? If so, standard human interaction doesn't apply to girls, and they're all different and don't have a fucking clue what they want most of the time. Not even signals will help you.
When I'm simply physically attracted to some girl I meet somewhere where inhibitions are low (club, party etc), I'm a lot more likely to make a move, and things work out for the better. But if it's a girl I meet normally and get to know her after hanging out a few times, and feel an emotional attraction... well then I'm fucked.
I guess all I can say is that I hope you have better luck than I do.

Why do you need to ask her out?

Ask her to a movie, or shopping or something away from school so you can talk away from her's and your friends for a while.

If all goes well, you will be unofficially going out.