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Of course I want to fuck the hell out of her haha. Maybe even start a relationship with her. I just never learned how to "hang out." What goes on in a "hang out?" Should there be other friends? What should we go do prompts that "hang out?"
Why do you entertain the though of a relationship with someone you don't even know?
Aside from that you met her yesterday and invited her to your dorm today. There's a difference between having a woman somewhat interested in you, and having her down to fuck you.
She's not stupid, she knows exactly what "teaching her chess" means.
Be lighter next time. If you want things to get sexual you need a basis for that. You need to escalate. You need sexual tension. You can't just jump from point A to point Z.
Invite her for an ice cream. A walk. A coffee. Then you can attempt to move things to your dorm for cat videos.