They have a great atmosphere and humor (back when games didn't take themselves too seriously), but the puzzles and gameplay might be frustrating by today's standards. Use a walkthrough if needed. All the games are on GOG, with remastered versions of 1 and 2 available.
The remastered versions are a good start, agreed.
@Stg-Flame: This is by far the easiest (and also cheapest) way of getting to know them and see whether the game's humor is up your alley. And you can switch between remastered and original graphics (well, SVGA for TSOMI, but close enough...).
You also get the bonus of a full voice-over, with some fiddling around even for the original version.
Here, you can find my fan-made unofficial patches, including the Monkey Island Ultimate Talkie Edition builders, and the Deponia audio fixes.
gratissaugen.de
Here, you can find my fan-made unofficial patches, including the Monkey Island Ultimate Talkie Edition builders, and the Deponia audio fixes.
gratissaugen.de
Also, if you're really into it, you could go full-on MIDI goodness, get a Roland MT-32 or CM-32L (plus MIDI interface if you don't have one) and listen to these great tunes as they were intended

But that's optional (although very much recommended by me

). Nowadays, I use DOSBox and the files from my original CD-ROMs and floppy disks. Plus a Roland UM-ONE Mk. II as a MIDI interface, connecting DOSBox to my stack of MIDI devices. But that's a lot more fiddling around and I digress.
If you are at least slightly interested in funny, story-driven games, without any time-pressure (well almost, but I won't spoil it

) or fear of dead-ends and virtual death, and have the time (you'll also need pen & paper to take some notes, unless you have a very good memory

), get them now, or when they go on sale.
These are one of my most beloved games, and I do replay them from time to time, although I know all the puzzle solutions by heart, after all those years

It's just Grim Fandango that I like even more.