Wow, who resurrected this excellent thread? 2002! What a great year for, um, something or other. Wait, wasn't that the year that Sa... (shuts mouth quickly in case he's right)
My first was no foolin' Monkey Island 2: LeChuck's Revenge. I was 9, it was on the Amiga, and I cheated all the way through. I borrowed it from a mate who had one part of a walkthrough and I had the second part.
The first adventure I'd play properly was Day of the Tentacle in 1997, which started me on the slippery slope of LucasArts Adventures.
My first non-LucasArts adventure was Discworld. I got stuck on the first screen. It would take me years to finish it. Discworld Noir on the other hand was the first I actually liked.