Ahh...The Point 'n' Clicker Adventure. My bread and butter when growing up.
For someone new to the Genre, I'll go with what Llil said, and recommend Gemini Rue. For me, the story, atmosphere, and soundtrack were incredibly immersive. And since the puzzles were completely logical, it keeps a nice fast pace going. So if you like Cyberpunk stuff such as Bladerunner, I think you'd like Gemini Rue.
An incredibly unique one would have to be Full Throttle, by LucasArts. I've never seen anything like it before, or since.
It's not too hard, has some fantastic animation, a cool story, and great dialog. Basically you play as the leader of a biker gang clinging to the old ways in a somewhat futuristic earth. The trailer really does it justice...
Once you have a bit of experience under your belt, you might want to consider playing the holy grail of Adventure games: Indiana Jones and the Fate of Atlantis. I played this a lot in my childhood, and I loved it. But I recently went back to replay it just a few months ago to see how it holds up today, and I was really quite amazed...I actually enjoyed even more than when I was a kid. And not like "Oh man, that's a nice hit of nostalgia", but more "Holy Fuzzsticks, this is better than the movies!". The setting, the plot, the places you go...It all culminates into a thrilling globetrotting adventure, worthy of a book or movie!
It's also very replayable. As there are 3 ways to solve everything. So all in all, highly recommend it!
Honorable mentions:
Lost Horizon: Globetrotting adventure, much like Indiana Jones, quite excellent.
Primordia: Published by the same people who did Gemini Rue, Wadjeteye (In fact, all of their stuff is good, check it out). Dystopian Future where only robots remain. Very cool artwork and story.
The Dig: LucasArts went all out on this, directed by Steven Speilberg, about 3 people stranded on an alien planet
Sierra games: Basically just take your pick on what interests you, they're all pretty decent. Space Quest and Quest for Glory, in particular.
Monkey Island Series: Pretty fun comical adventures, some puzzles can be a bit hard though.
Dreamweb: Dystopian cyberpunk future, very moody atmosphere. Has it's problems, but I loved it.