You forced an IOI. Immediately you set yourself apart from everyone else by making contact with her and keeping your frame as you did it. I do this all the time now and it works really well on confident and attractive women. In those cases you can let them come to you.
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This girls is 9/10 intelligent and interesting. I would really like to have a relationship with her, but dating is my sticking point. I have no idea where to take her or what to talk to her about. In general it is extremely hard for me to keep a conversations with girls on dates and usually I end up being dead silent or even awkward. Also worth to mention, this girl and me have literally opposite interests and very different personalities. She is very straightforward and blunt.
It doesn't matter if you have similar interests. If you're passionate about something and can talk about it without boring yourself she'll be interested. Don't put so much pressure on talking and don't worry about where to bring her. Be unconventional, do whatever and act like it's not a big deal to go where you take her.
She wants a date? Fine. Bring her to a bar for a drink or two. She said she didn't want alcohol involved but that just means she doesn't want to be fucking hammered. This chick wants a relationship. One or two drinks is fine. If you don't want to do that just bring her for a walk or something. Take a walk, talk, share funny stories and don't be afraid to ask her about herself. It's day 2, you can ask about her and let her do all the talking.