I recently met a girl who goes to my school while on a flight. We had a decent conversation so before we got off I told her that I didn't know many people at our university and that if she was cool with it I'd like to keep in touch with her. She was nice about it and agreed and told me to contact her after summer as we both will be far from university this break. My question is how do I turn this acquaintance into a friend? My original thought is to just invite her to a comedy show or something. I just don't know if that is a bit much for someone you just met. I guess I could invite her to get coffee so I could get to know her more but I didn't want it to seem like I was trying to actually get with her. There's nothing wrong with dating the same sex I just didn't want to mess up a friendship before it started because she thought I was inviting her on a date and not to stereotype but I think she might be Muslim so I don't want her get freaked out. I haven't really been around people that much but I know I am going to start working soon so I'd like to know if I am over thinking the coffee thing or what I should invite her to do. Any suggestions for this girl or future friends?

*i posted this in the school section first on accident but I think it is best placed here*