I know what you mean - went through it myself when I was at school. If you want to appear as normal as possible in front of her, the worst thing you could do is to concentrate too much on appearing normal. I got over my fear of this girl by throwing caution to the wind and invading the tennis court where she was playing netball in. I totally disrupted the whole game by involving enthusiastically myself with all the grace of a sheep. She said 'yes' before I was dragged from the court by Mr Hawley, though.
There is no easy answer. You're just gonna have to marshal your courage, bite the bullet and just....go for it.
Talk about a story from class or something in the news for an icebreaker...then talk about something like a movie that's playing...
when you start talking keep it simple and keep you body angled away from her this will make you seem uninterested but she will psychologically crave your attention from this point because of your body language
Find something you two can do together and enjoy. Be blunt about wanting to go out and do that on Valentines day however. If you can get along well doing something you both enjoy then it will be more about having fun and being happy than pure interest.
It is not really strategics. You ask a question and work from there. Let it happen naturally and stop worrying about it so much if you honestly believe thinking too much into it is a bad thing.