Allow me to second this. In my experience, which is extensive in the world of internet dating, keep it public at first.
You want to pick places that are relatively quiet, places where you can talk, and places with alot of stimulus, audio, visual, etc. Sidewalk cafe type places are great for this. You want to have lots of conversation triggers, look at that neat car, why does that guy have blue/green striped hair, etc.
If she is requesting the second get together, good sign. Second and third dates are where things get emotionally messy. Unfortunately you are on your own for interpretation of posture, body language, etc. In general since you have a second date, and you want there to be a third, you have to pay attention.
One of the things I always used as kinda a litmus test was superficial physical affection. I went to great lengths to respect personal space on first meets except for hug/kiss on cheek at parting. First meets off the net are for chemistry checks, I never really considered them “dates”. IF she seem receptive to a hug, maybe even snuggles in a bit, great, she at least sees you as safe if not attractive in some way. Look for any opportunity to touch her, breifly and in an appropriate fashion, offer your hand to help her out of the car or a seat at a restaraunt, help her with her coat, etc. I would often encounter 10-15 such opportunities in a typical dinner and a movie type situation. You want her to become accustomed to the fact that you can touch her without it being a sex act and that maybe you actually care about something beyond dipping your wick.
In general you will probably need to tell us a bit more about this girl. What DOES she like to do, where has she been, what do you know about her life. Everyone is different but there are almost always some basic psych and family dynamic issues that can be helpful in determining how to help you score points or mines to avoid.
Rule#1 is always be yourself, and be on your best behavior. She is probably assuming you are and if you are an obnoxious slob now whats it going to be like 10 years and 3 kids later.