Like Oh Coy! said start with Dirt. That is easily the best AIC album IMO. Every song on that CD is a classic as far as I am concerned and it's one of the best rock albums ever made...period. The whole album is very dark and depressing. Most of the songs have to do with Layne Staley's herion addiction ("Sickman", "Junkhead", "Dirt," "God Smack," "Hate To Feel" and "Angry Chair.") Dirt also has one of my favorite songs in any genre of music ever..."Would?" To this day that song still gives me chills when I hear it.
After Dirt I'd pick up Jar of Flies. It's a shorter album (7 songs) but it is an amazing album. Like a few people already said "Nutshell" is a great fucking song. It also has "I Stay Away" and "Rotten Apple" on it which are another two of my favorite AIC songs (I have a lot of favorite AIC songs if you can't tell lol.)
Once you get both Dirt and Jar of Flies I would say you need to go get their MTV Unplugged album. I really love this album a lot. It's really cool to hear some of their heavy songs done acousticly. It's sad though because it's one of the last performances they gave before Layne became a herion hermit. You can tell his voice is a little weaker from all the drugs and I don't like saying it but because it's acoustic it fits right in with the softer feel of the music. Great fucking album.
Next I'd pick up their Sap EP. It's called Sap due to its "sappy", or softer, sound than the bands previous album, Facelift. There is five songs total (Love Song is unlisted) and it's a great little EP. The stand out tracks to me are "Brother" and "Got Me Wrong." All the songs on the album are great though.
Then I'd get their last proper album the self titled Alice In Chains record. This album features "Heaven Beside You" which just is an amazing fucking song. Other noteables on the album are IMO "Grind", "Head Creeps", "Again", "God Am", and "Frogs"
After that get first full length release Facelift. It still has a little of a raw sound and the band isn't as tight musicly as they would become...it's still a really good album though. "We Die Young", "Man in the Box", "Sea of Sorrow", and "Love, Hate, Love" are my favorites of the album.
Lastly I'd get their Live album. It's a really good live album that covers all their classics and even one of their oldest songs "Queen of the Rodeo" which is also on Music Bank though.
Besides those they have the We Die Young EP that's three songs long but all but one of those ("Killing Yourself") is on Facelift. Then there is Music Bank which is three CD's and a fourth CD-ROM disc. It's a great box set and has a really interesting book that accompanies it. It also has a bunch of unreleased demos (the Rooster demo is fucking amazing IMO.), a few unreleased songs, and a few remixs. Nothing Safe: Best of the Box is like an abridged version of their box set Music Bank. Then they have just a Greatest Hits album which is just that...a greatest hits album. It's the only album from AIC I don't own because I got everything on it.
If you decide that Layne Staley has one of the greatest voices in rock ever (like a lot of us do) check out his side project Mad Season. They only had one album called Above but it is a classic from start to finish. It has some of Layne's best vocals and is a little more laid back then AIC so his vocals really show through. It's sort of a northwest grunge super group seeing that Mike McCready (guitar) of Pearl Jam and Barrett Martin (drums) of Screaming Trees are also in the group. Mark Lanegan of Screaming Tree's also does some singing on the album.
My comp has been acting up so it would take forever to upload some songs or I would. I'm sure Ricky will be by to drop some albums for download sooner or later though.
Well there you have it. As you can tell Alice In Chains are one of if not my favorite rock bands ever...fuck they are most likely my favorite band/group/artist in any genre of music. My list changes all the time but my top three in rock are always...in no order...Alice In Chains, Tool, and Joy Division.