How should I get to know this girl?

I am a freshman in college. This girl I'm interested in is also a freshman. We are both aviation students. In April of last year I posted in the Facebook group for the class of 2016 if there were any aviation students out there. She commented "yeah buddy!" on that, and then I added her as a friend.

I was at a dining hall once and I saw her, she was looking at me for a few seconds. She had a boyfriend for a month but he dumped her.

For the spring semester I moved to a different residence hall, which is the main freshman building with 700 people. There was one time when I was going inside the building and she was behind me along with other people. I scanned my card to unlock the door and held it open for her and everyone else. She said "thank you" real nicely.

There was one day this week where I was walking out of the building and she was walking into the building and I held the door open for her, and she said "thank you" nicely again. Just today I was waiting at a bus stop and then she walked up, but I isn't notice that it was her until the bus arrived and then she turned around. I wasn't expecting to see her there but she was looking at me, and I quickly got on the bus because I was nervous. On the bus it felt like she was looking at me, although I don't really know.

So the thing is I'm really shy/passive. I've never had a girlfriend before. I think she is a bit shy too but I don't definitively know. She has a bunch of good friends. She is really funny on twitter. We have some things in common (obviously the aviation interest) but we also have some similar music and TV show interests.

I think she may not know who I am really. She doesn't follow me on twitter although I follow her. She said something before how she doesn't like people just randomly chatting her up on Facebook or whatever. Other than one boyfriend that she had for a month, who dumped her about a month ago, she has never had a boyfriend.

My question is how should I try to get to know her? She's pretty cute actually and as far as I know she doesn't really have guys over her or anything. So what do you suggest? Thank you for your help.
How should I get to know this girl?
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