I've been spending the last couple years watching all the Star Trek episodes and movies in their chronological order. I remember seeing a couple of the Next Generation series shows growing up, but I never could get into the original series at that time. It didn't make sense.
I'd have to say that my favorite crew so far (I haven't gotten to DS9 or Voyager) is the Next Generation crew. Maybe it's because they're the ones from my childhood...or maybe it's the renewed vision in that show which projects a positive future void of material desires, or the mission of peace.
Do you have a favorite Star Trek crew/series?