It depends, but generally you should exchange a few messages over the website you're using, and if you hit it off, start texting. I'd say for most people a couple weeks of texting is about average.
While I definitely agree with Aaron, it also kind of goes partially with your instinct. I was talking to this one girl online, she asked me out probably within a week and we have a date for Wednesday, this was before we even exchanged numbers. Sometimes it's worth taking the jump, if you feel comfortable enough.
I had a girl message me once basically saying, "Hi, I liked looking at your profile, would you like to meet up for coffee to get to know each other?" Uhh, hold your horses woman lol. She should have attempted to get to know me more through the website before asking to go for coffee. A couple of weeks sounds reasonable.
Sometimes yes, sometimes no. It all depends on the person and how well you click, so to say. I know, not terribly helpful, i'd say err on the safe side. If it's going REALLY well, maybe go a little early. If there is constant chat and flirting and you just seem to get along, next step doesn't need to wait as long IMHO. I'm also kind of hopelessly single, so we'll see.
My fiancee and I met online. We talked back and forth through the website for about 3 months before agreeing to meet in a public place. After we met, we then exchanged numbers and facebook information...and well...now, a year and a half later, we're getting married lol
That's awesome Did you have to go though a lot of dates with other people before meeting your future fiancee?