Gemini basically is the sign of communication and curiosity. So if you've just met a Gemini and you're interested, the first thing is to be a good conversationalist. This doesn't mean that you should do all the talking, because certainly Gemini is very willing to talk and talk and talk about any old thing. Therefore, you should be a good listener.
The whole point of conversation is give and take. Do have something interesting to say. Do have something about your family, your background, where you've traveled. You can just bring that up in conversation, because that is what immediately will attract a Gemini.
Gemini's curiosity is also piqued by discovery. Therefore, if you can in a way slowly reveal things about yourself, not necessarily very personal right away. The sense of discovery is something that keeps Gemini hooked. Gemini is also one that loves to experiment. Of course, this can run the gamut from experimenting with going to a new art gallery or going to a new place that's just opened, trying out a new food.
It is a sense that there's something new to be discovered, and here is someone who is willing, happy and able to do it with him or her. Essentially, the key to attracting, moving closer to, and keeping a Gemini is to be a good friend.