Okay, If you have started with Nighthawks, then you are digging the jazzy beat poetry stuff.
Tom's releases can be split in two halves, 1973-82 jazzy, blues, beat stuff.
1983-02 much more experimental, but still very jazz blues influenced.
Here is my take on Tom's albums.
Closing Time Asylum 1973:
Great debut, some lovely songs. Tom was finding his feet with this album.
Standout tracks include: Ol' 55, I Hope That I Don't Fall in Love with You, Grapefruit Moon.
Recommended.
The Heart of Saturday Night Asylum 1974:
The first real classic Waits sounding album. The title track is a Waits classic. Standout tracks include: New Coat of Paint, San Diego Serenade, The Heart of Saturday Night.
Recommended.
Nighthawks at the Diner Asylum 1975:
You have this album, not one of my faves, but a great live look at Tom in 75.
Small Change Asylum 1976:
The first essential Tom album, a total classic. Standout tracks include: Tom Traubert's Blues, Step Right Up, I Wish I Was in New Orleans, Invitation to the Blues, Bad Liver and A Broken Heart .
Highly Recommended.
Foreign Affairs Asylum 1977:
Not quite the great album it should have been, but it sees Tom moving in the right direction.
Standout tracks: Muriel, A Sight For Sore Eyes.
Blue Valentine Asylum 1978:
Right back on track, this is probably the last "old style" Waits album. Contains some truly great songs.
Standout tracks: Christmas Card From A Hooker In Minneapolis, Romeo Is Bleeding, Blue Valentines.
Recommended.
One from the Heart Original Soundtrack Recording CBS 1982:
The soundtrack to the Francis Ford Coppolla 82 flop. This is a really nice album, probably more for big Waits fans. Most of the songs are co-sung with Crystal Gale.
Standout tracks: Old Boyfriends, Broken Bicycles.
Recommended.
Swordfishtrombones Island 1983:
Now ya' talking! This is where Waits truly became a master.
Songs about midgets, murder, whores, insects, it's all here.
Standout tracks: Underground, Shore Leave, Dave The Butcher (Instrumental), Johnsburg Illinois, Frank's Wild Years, Swordfishtrombones, Soldier's Things, Gin Soaked Boy, Trouble's Braids
Highly Recommended.
Rain Dogs Island 1985:
What can I say, just when you think Tom could not improve on Swordfish, he unleashes Rain dogs on us. Unbelievable GENUIS.
Standout tracks: pretty much the whole album!
Highly Recommended.
Frank's Wild Years Island 1987
Quite close to the previous two albums. A real grower, almost as good as Rain Dogs.
Standout tracks: Most.
Highly Recommended.
Big Time Island 1988
Live album, nice but for Big Tom fans only.
Night on Earth Original Soundtrack Recording Island 1992
Mostly instrumental soundtrack album, nice, but again for fans only.
Bone Machine Island 1992
Tom first proper studio album in Five years. And it was well worth the wait.
It seems Tom had been having some very dark nightmares.
This album is very dark. One of his best.
Standout tracks: most.
Highly Recommended.
The Black Rider Island 1993
From a German play Tom wrote with William Burrows.
Full of carnivals and Skeletons. Not for everyone, but I love it.
Recommended.
Mule Variations Epitaph 1999
Big gap in studio albums again. This album sees Tom in a very bluesy mood. Lot's of very Scratchy old sounding songs.
Blood Money Anti 2002
Great album which is basically a re-recording by Tom of the songs he wrote for the Robert Wilson play of Woyzeck.
Highly Recommended.
Alice Anti 2002
Another Play. This album spent many years as a bootleg. Tom re-recorded it at the same time as Blood Money. Classic.
Highly Recommended.
I think the best places to start are(in order of release), Heart of Saturday Night, Small Change, Heart attack and Vine, Swordfishtrombones, Rain Dogs, Bone Machine. Then but em' all!!
There are many compilations around also.