I am 31. I started watching TNG in '91 or '92 so I was a "bit late" to the party, though I remeber seeing episodes here-and-there before then as my parents watched the occasional episode in our home, but neither of them are real fans of the series but they'd catch the occasional "heavy" episode, I remember them watching "The Offspring" when it first aired.
TNG is what brought me into the franchise the first episode I watched on my own volition was, I believe the S-5 episode "Cause and Effect" and then I was hooked as I got pulled more and more into the series.
Until then I thought it was typical Sci-Fi nonsense, aliens, fighting, space battles, bah! But when I discovered it was about exploration, discovery and humanity I was awed. Got more into TNG, catching older episodes in second-run syndication and watching new episodes as they came, tried to get into DS9 when it premiered but it failed to "hook" me when it started off I'd only watch it occasionaly after that. Voyager I watched fairly reguarly when it started until I realized it wasn't taking its own premise seriously and I pretty much detatched from it, again catching it on occasion. Enterprise I watched all the way through and TOS I watched on occasion when it was in syndication but didn't really watch it until the DVD Box-Sets came out a few years ago.