I didn't watch TNG during it's first run. At the time, I didn't want to watch anything other than the original Star Trek. I wasn't a huge sci-fi fan so I'm surprised I'd be so picky but I was. It wasn't until about the year 2000 that I was up late and nothing else was on. I was flipping the channels. TNG was on. What caught my attention and got me to put down the remote was they had just beamed aboard a metal object that had "NASA" written on it. I thought I'd watch a bit. That was "The Royale" episode where they discover what happened to that missing astronaut. When they find out his identity, Riker says, "Rest in peace, colonel". It's like bringing a missing soldier back home.
I've thought about it several times that this episode particularly was what I needed to get me to be a fan. I'm partial to stories like this where someone lost is "brought home" sort of. They don't actually bring his body home, but the knowledge of what happened to him. The other parts of the story, like the mobsters, was silly. Many people don't like this episode, but this story got me to watch. AND, Riker was especially handsome so I watched a few more episodes. After that, I was a fan. Luckily at that time there were a couple of networks that aired 3 or 4 episodes a day. I was able to catch up with the series quickly. Became a Picard fan, too.
I didn't like DS9 much. I agree with that other poster that it didn't really seem like a Trek show. It was just aliens in outer space. It never clicked with me. When TNG wasn't on, I'd watch Voyager. It was okay, but TNG is far and away my favorite of ALL the Star Trek series.