It was the late 90s, which was when I had my first computer. My brother had bought "Titanic: Adventure out of Time" and didn't really care for it. He gave it to me to try, and that's what started it all. I had never known anything like it, and I loved it. That game is still special to me, as a result. I still have the box.
Titanic: Adventure Out of Time was my 4th adventure game, after Myst, Shivers, and Shivers 2. Titanic was the first adventure game I played that had conversation topics you could choose from. At the time I was impressed by that.
I got my first computer in 1996, a Compaq Win 95. I got Myst for free, bundled with the printer. So, I had to try it. I didn't like it at all, probably because I didn't understood what was being said. About the same time I bought Fate of Atlantis, and that is still one of my favorite games.
Fate of Atlantis is a good one. I had to order my copy online, and I got it direct from the LucasArts Company store, as part of the LucasArts Archives Vol. I collection. I wish I'd ordered more games from them while they were still around.