It depends on whether she's in your social circle (e.g. group of friends, classroom, work, etc.). I believe it's better to not dip into your social circle, so I focus on meeting new, random women. Here's how I ask these women out:
Keep things light and casual so it's not a big, scary thing. This makes it easy for her to commit, as well as keeping it low pressure for the two of you.
With that in mind, ask her if she'd like to grab a drink or coffee (pick the latter if she doesn't drink alcohol). Some people advise not asking, which I've done in the past, but I think asking is not only more socially intelligent, it also gets her to invest and it feels like she's part of the decision making process, both of which she'll appreciate.
If she agrees to grab a drink or coffee, then tell her to exchange phone numbers by pulling out your cell phone. Don't cop out with Facebook or business cards. Those things are impersonal and she can get the wrong idea.
Once you've exchanged phone numbers, right away text her that you're glad to have met her and tell her to save your number.
In a few days, follow up with her on that drink or coffee and ask her what her schedule is like for the rest of the week. From there, agree on a day, time, and venue.
That simple. Don't get trapped into spending hours texting. It's best to keep texting to coordinate logistics. Only in special circumstances should you exchange "fun" text messages. It's too difficult to build a connection over text. Save the fun and connection stuff when you meet her in person.
If you want to ask someone out in your social circle, I don't have much experience with that. But I can say it's ideal to date her away from the rest of the group. You want to make it private. This can be done by planning a date in advanced or on the fly when you're hanging out with the group, and you pull her aside for a "detour."