Shadowgate is a great game! There are two more games in the series, and a fourth is in development. Sadly, I have only played the NES game(awesome game!), which is like a darked King's Quest game in first person view. But the others look great as well!

I'm a big fan of point & click games, myself. Here are some suggestions for games that, in my personal opinion are a must:

The Longest Journey - I'm not talking about Dreamfall: The Longest Journey, but the original game from 1998. It is one of the most epic stories I've ever seen in a game ever. Starts out a bit slow, but soon picks up the pace!
Still Life - A great, modern style point & click game where you play as an FBI agent tracking a serial killer. Dark, and I'd say it's probably one of the best in the crime thriller category. Sadly, the sequel didn't do it justice.
Day of the Tentacle - This is just a crazy classic that shouldn't be missed, despite the strangeness of the title! Made by the same guys who created the original Monkey Island games. Tons of fun!
King's Quest VI: Heir Today, Gone Tomorrow - In my opinion the best KQ game by far. Might've had to do with the fact that Jane Jensen also worked on this. Speaking of which ...
Gabriel Knight - All three games here are worth playing. Seriously good games, but all are very different from each other. The first is a classic 2D adventure, the second is an FMV game, and the third is in rather dated 3D with some rather unorthodox controls(though still P&C). But they are all dark, mysterious supernatural thrillers with great stories!
Scratches - This is an incredibly good first person view point & click game! It's rather short, but I highly advise playing through it at least once. In the dark. At night.

That's what I can remember of those that I would say you shouldn't go without at least trying. Some other suggestions:
Beneath a Steel Sky - Dystopian sci-fi thriller. Overall great game! This is also free on GOG.com.
Grim Fandango - By the Monkey Island guys again. Great game!
Syberia - Great modern adventure! There are two games in the series, and they should be played in succession as they are more like two halves of the same story.
The Moment of Silence - If you look away from some annoying glitches, this is a great game! Dark sci-fi storyline.
The Dig - Another classic. Very dark sci-fi.
Cognition: An Erica Reed Thriller - Another really good crime thriller, this one a bit darker than Still Life, and with supernatural elements. It's in episodic format, and the fourth and last episode will probably be out within a month or two.
The White Chamber - Sci-fi horror game. Very cool!
Gray Matter - Another game by Jane Jensen. It's incredibly good, but forgets to explain a few of the plot elements that ought to have been explained.
Barrow Hill - First person horror mystery. A bit short, but it's full of atmosphere and well worth playing!
The Chzo Mythos - A horror series created by Yahtzee, of Zero Punctuation fame. The games in the series are 5 Days a Stranger, 7 Days a Skeptic, Trilby's Notes and 6 Days a Sacrifice, and they are well worth playing!


Okay, that's it for now. If I remember any other games I would highly recommend, I'll post them.