I'd say go with Firefly and then Serenity. If she doesn't like that, there's no way she is going to like Star Trek.
Also, very little Star Trek stands the test of time. So I suggest the 2009 movie which is easily the best thing to ever come out of the Star Trek franchise (though traditionalists might not like to admit it). It's a great movie.
I concur. That's what I tell people: "Go to a big brand retail store and see which brand you like, then I'll look for the best model/spec/price combination for you online.