Most Helpful Girl
Open up and tell something embarrassing or goofy (but not too bad) about yourself. Showing that you're human and make mistakes too is always reassuring. Or maybe admit that you collect kitten calendars or that you're a major star wars dork. These aren't very "cozy", but if they work you can gradually branch out and get into deeper conversations the more you get to know each other. Another thing is choose a comfortable setting. Alone in your house might be a bit much at first, but maybe a movie or some interesting activity would be fine. Nothing too serious too soon.