Interesting/Confusing situation?

For a few months now I've been talking off and on with a girl at my gym. For a month we just exchanged eye contact then I manned up and struck a conversation with her. I asked her if she graduated from my high school and told her she looked familiar. She looked confused at first and awkward at first but she started lightening up with a big grin. She asked where I went and so forth and asked what my name was. The conversation eventually reached an end and she walked off with a big grin. We spoke a few more times very briefly..but she always had a smile. Most of the time she would walk by me with her face buried in her phone. There was a period of time where I didn't see her and when I finally did I joked around and asked where she had been. She smiled and told me she was out of town. My friend who works there said she had been signing in everyday. I thought, why would she lie about that? I've seen her a few more times since then. I've been a little discouraged and haven't talked to her, but she made eye contact with me several times, and also got on a tread mill very close to me the other night. My friend who works at the gym is actually dating her cousin. He spoke to his girlfriend about the situation in hopes to "expedite" the situation. His girlfriend hinted that I was wanting to ask her out and get to know her more, but didn't mention who it was. She knew right off the bat who it was. He told me that she said I seemed really shy. I asked him if she found me attractive and he wouldn't tell me anything more. He just kept telling me next time you see her talk to her. I tried to dig as much information as I could out of him but he just kept pushing me to talk to her. I want to find out additional information because if she doesn't find me attractive..why bother? I rarely see her, so at this point should I send her a friend request on facebook and said a casual, "hey, how are you" message? I don't want to waste anymore time with this, so I just want to get it over with. I want to talk to her a few times. ask her out, if she says no then it's over. If she says yes then I'm money. I do like her because she really is a good girl that has her priorities straight from what I've learned and very cute as well. It's so hard to strike a conversation with her because she seems very shy and conserved and that makes me nervous. What should I do?