I'm voting 3rd party. You're telling me that's a vote for Romney and Romney supporters tell me that it's a vote for Obama. So, by everyone's count I've got three votes now. That sounds pretty awesome to me.
TNG, but the first season is pretty slow so show some of the better ones at first like Darmok, Ship in a Bottle, or the ones with Q. If she likes it go back to the beginning and see if she can follow from there.
After that usually DS9, then it's really it's personal preference from there. TOS is good, but it's really dated so a lot of people actually skip it or watch it last.
It really hit home how sad this was when I was watching Stargate SG-1 and they had to send up a space shuttle to get Jack O'Neil and company out of a stranded alien space craft in orbit. I then realized in current times, the use of a space shuttle would be part of the science fiction.