This is at once a very easy and very difficult question to answer, but let's give it a whirl! (Disclaimer: This method is not one-size-fits-all. Your mileage will vary.)

You write a first contact email. It's short, sweet, personalized to their profile, and ends with a question for them to answer. They write you back to say, "Thanks, but no thanks." Do not pass go, do not collect date.

You write a first contact email. It's short, sweet, personalized to their profile, and ends with a question for them to answer. They write you back and answer your question, but do not ask any of their own. Write back, continue to show interest, but do not pass go, do not collect date (yet).

You receive a first contact email. You enjoy what you see and read, so you reply back with an email that is short, sweet, personalized to their profile, and ends with a question for them to answer. (Seeing the pattern here?)

You receive a second email from the same person. It is short, perfunctory, and not terribly well thought-out. Write back, continue to show interest, but do not pass go, do not collect date (yet).

You receive a second email from the same person. They ask more questions about you and reveal more about their life. Write back, continue to show interest, offer your IM username(s) and phone number, should they wish to use them.

You receive a third email from the same person, who has not yet used your IMs or phone number. They ask more questions about you and reveal more about their life. Write back, continue to show interest, ask if they would like to chat via IMs or phone sometime.

You begin to chat with the person via IMs or phone. At this point, you should start to get the idea of whether or not you have the intellectual and social chemistry to warrant meeting in real life. Pass go, bring up the idea of meeting up sometime in the near future.

You receive a fourth email from the same person, who has not yet used your IMs or phone number. They ask more questions about you and reveal more about their life. Now is the time to bring up the idea of going on a date.

The important thing is not to waste time chatting back and forth with someone who doesn't want to take things to the next level and meet in real life. There are (sadly) folks out there who use online dating as a way to accrue fun chat buddies or people to fawn all over them every time they log in to the internet. The only way to tell if your love interest is serious is if they accept to meet out in the real world.

Now, there are folks who claim that asking someone out on a date in a first or second email is the way to go. LADIES - this will probably work for you. GUYS - more often than not, this will not work for you. Is it sexist? Sure is. Is it how it is out there in the real world? Yep, deal with it. More often than not, guys, if you ask a girl out in a first email, she's going to assume you do that for everyone and that she's not that special. Take the time to get to know her and show her she means something to you and she'll be more receptive to that first date offer.