Here is a listing of the first performances of some of the Dead's early songs. I skipped covers for the most part, but thought a simple list of first versions would show some interesting changes in how the songs developed, if anyone wants to go through and compare.
The initial idea was to see which tunes improved from their first performance to the album versions; but there's no general answer - each song has its own history.
Some were excellent the first time out; others took a long time to settle into place; some were played for months before being recorded; others popped onto the stage fresh from studio rehearsals. Since Dead songs often took years to reach their full potential, it's rare to find a studio version that's the "standard" for a song. Early versions are usually rather bare and rough-edged, which makes them interesting to hear since our ears tend to fill in what's missing.
I may have skipped some songs inadvertently, so add a comment if any are missing. This list goes up to '77, and is mostly limited to the Dead's original songs that appeared on albums (so a lot of their unreleased '66 songs are not included), and a few released covers.
http://www.archive.org/details/gd66-01-xx.sbd.hanno-uli.18846.sbeok.shnf - Viola Lee Blues studio rehearsal.
For most of the songs on the Dead's first two albums, we don't have "first" performances since so many shows are missing. 1967 was a year of nonstop rehearsals, so early versions of Anthem tunes like the Other One & New Potato are very similar to the album recordings. Viola Lee's unique in that we have the rehearsal BEFORE the first live version.
http://www.archive.org/details/gd66-02-25.sbd.unknown.1593.sbefail.shnf - Never released in a studio version, but this list wouldn't be complete without mentioning our earliest live I Know You Rider (which is similar to the Nov '65 studio demo).
As for some of the other songs on the first album, I'll just list a few -
Our first Morning Dew comes from the same month the album was recorded -
Good Morning Little Schoolgirl was one of Pigpen's earliest numbers, and in fact our first Schoolgirl from early '66 is also the first Dead medley (sandwiched with You Don't Love Me) -
http://www.archive.org/details/gd1966-XX-XX.sbd.GEMS.81254.flac16 (This set also has the earliest live Viola Lees.)
Cream Puff War was played throughout '66, and this one's the earliest (there's another early one on the Rare Cuts CD) -
Golden Road, unfortunately, we only have a couple performances from after the album release -
http://www.archive.org/details/gd1966-01-08.sbd.bershaw.5410.shnf - our first live Caution (though there's a short studio cut from November '65). (This show also has our first Death Don't Have No Mercy, which never changed much.)
http://www.archive.org/details/gd67-06-18.sbd.mandrake.19586.sbeok.shnf - a primordial Alligator.
http://www.archive.org/details/gd67-08-04.sbd.hanno.16752.sbeok.shnf - I think this is the earliest New Potato we have (though some are misdated earlier).
http://www.archive.org/details/gd67-08-05.sbd.hanno.16753.sbeok.shnf - Lovelight. I know, this song was only done live, but these early ones are different.
http://www.archive.org/details/gd67-10-22.sbd.miller.18101.sbeok.shnf - first Other One, different lyrics.
http://www.archive.org/details/gd1968-01-17.sbd.cotsman.11795.shnf - first live Dark Star, China Cat, Eleven, Born Cross-Eyed.
Dark Star had already been recorded for the single, so this is an example of a live debut that's close to the studio version but sounds very undeveloped since they kept expanding it for years.
China Cat, on the other hand, wouldn't be recorded for another year, and these early '68 versions are not only in a different key (so Garcia sings strangely) but are paired with the Eleven.
http://www.archive.org/details/gd68-xx-xx.sbd.vernon.9426.sbeok.shnf - early St Stephens.
We're fortunate to have a few Stephens from June '68, shortly after its debut. 6/14 is the best, but these are in sbd quality. Many differences from later Stephens!
Most of the other Aoxomoxoa debuts don't sound too different from the album versions, since they were played at the same time they were recorded.
But for the record:
Dupree's (with harmonica) & Doin' That Rag -
http://www.archive.org/details/gd1968-12-07.sbd.miller.88674.sbeok.flac16 (only one)
Mountains of the Moon -
Cosmic Charlie -
http://www.archive.org/details/gd68-10-08.sbd.belaff.17691.sbeok.shnf (Hartbeats, great!) +
(The 4/26/69 encore of What's Become of the Baby doesn't count since I believe Bear was playing back the studio version while the band played feedback.)
http://www.archive.org/details/gd1970-01-01.studio.smith.91324.sbeok.flac16 - studio demo in June '69, Jerry solo Casey Jones, Dire Wolf, High Time.
http://www.archive.org/details/gd69-06-07.sbd.kaplan.9074.sbeok.shnf - first live Dire Wolf. But a more interesting alternate early version (Jerry on Pedal steel, Weir on vocals) is here - http://www.archive.org/details/gd69-06-27.sbd.samaritano.20547.sbeok.shnf
http://www.archive.org/details/gd69-06-21.early-late.aud-sbd.cotsman.16334.sbeok.shnf - first live High Time. These early versions are quite wispy.
http://www.archive.org/details/gd69-06-22.sbd.kaplan.17892.sbeok.shnf - first live Casey Jones. The early versions have a neat jam into the song, and this one's the best.
http://www.archive.org/details/gd69-08-21.sbd.cotsman.13850.sbeok.shnf - first live Easy Wind. It would take them a while to work out the time changes!
http://www.archive.org/details/gd69-11-01.sbd.cotsman.6298.sbeok.shnf - the first Uncle John's instrumental.
11/8/69 had the first Cumberland Blues, which needed a bit more practice....
The first one on the Archive is: http://www.archive.org/details/gd69-11-15.sbd.tiedrich.9535.sbeok.shnf
http://www.archive.org/details/gd69-12-04.sbd.cotsman.9496.sbeok.shnf - first live Black Peter, and first Uncle John's with lyrics.
http://www.archive.org/details/gd69-12-19.sbd.hanno.9183.sbeok.shnf - first live Mason's Children - a song they decided not to release, though they recorded a studio demo.
http://www.archive.org/details/gd69-12-20.sbd.cotsman.6301.sbefail.shnf - first live New Speedway Boogie. The chorus singing is, um, quite different.
http://www.archive.org/details/gd1970-03-20.sbd.smith.99182.sbeok.flac16 - first live Friend of the Devil (with some different lyrics).
We're lucky to have a tape of the original Dawson/Hunter recording of Friend of the Devil too! (from fall '69) -
(This tape also has the Dead working out the Uncle John's instrumental.)
http://www.archive.org/details/gd70-04-03.sbd.cotsman.4283.sbefail.shnf - first live Candyman (acoustic).
http://www.archive.org/details/gd1970-05-14.sbd.warner-evans.28716.sbeok.flac16 - first live Attics. Some of these early versions show the Dead struggling with it....
http://www.archive.org/details/gd70-06-07.sbd.hollister.98.sbeok.shnf - first live Sugar Magnolia. Needed some more work!
http://www.archive.org/details/gd70-07-30.sbd.cotsman.17077.sbeok.shnf - first live To Lay Me Down (acoustic). Also, a unique Dead studio version from this summer was released on the So Many Roads box set.
http://www.archive.org/details/gd70-08-18.aud.yerys.1346.sbeok.shnf - first live Truckin' - acoustic, with Pigpen on piano. (Also the first Ripple, Brokedown Palace & Operator; they were recording American Beauty this month.)
http://www.archive.org/details/gd70-09-17.aud.warner.16090.sbeok.shnf - our only Box of Rain from 1970. According to Deadlists, Pigpen's on piano, Jerry's on pedal steel, and someone's on fiddle, but frankly it's nearly inaudible.
http://www.archive.org/details/gd70-09-18.sbd-aud.cotsman.17893.sbeok.shnf - our first live Till the Morning Comes (AUD). This song never changed much. More well-known versions came on 10/4 and 10/31/70.
Bertha & Bird Song were played at the 12/15/70 show with David Crosby, and Loser was played (with fiddle!) at the PERRO sessions, but they're not on the Archive.
http://www.archive.org/details/gd71-02-18.sbd.orf.107.sbeok.shnf - Famous show of course; first Dead versions of Bertha, Loser, Greatest Story, Wharf Rat, and Playing in the Band.
This show marks a change in the Dead's pattern: instead of dropping songs into the sets one-by-one when composed, now they tended to write a whole bunch of songs in a touring break and introduce them all at once.
http://www.archive.org/details/gd1971-02-19.aud.Vadnais.30230.flac16 - first Dead versions of Bird Song & Deal.
http://www.archive.org/details/gd71-07-31.winberg.weiner.5678.gdADT05.sbefail.shnf - first Sugaree and Mr. Charlie.
http://www.archive.org/details/gd71-10-19.orefm.clugston.3301.sbeok.shnf - A whole bunch of debuts. Clearly they had a busy month off!
Tennessee Jed, Jack Straw, Brown Eyed Women, Mexicali Blues, Comes A Time (with an extra verse), One More Saturday Night, Ramble On Rose
http://www.archive.org/details/gd72-03-05.sbd.miller.20739.sbeok.shnf - first Black Throated Wind.
http://www.archive.org/details/gd1972-03-21.sbd.miller.92395.sbeok.flac16 - first Looks Like Rain (w/ Pedal steel).
http://www.archive.org/details/gd72-04-17.sbd.vernon.9390.sbeok.shnf - the first He's Gone. Quick & snappy.
http://www.archive.org/details/gd72-06-17.aud.hanno.16470.sbeok.shnf - first Stella Blue (with Pigpen on organ) - poor AUD.
http://www.archive.org/details/gd72-07-16.sbd-aud.cotsman.11258.sbeok.shnf - first MS Half-Step (poor AUD).
(The next performance is more listenable - http://www.archive.org/details/gd1972-07-18.sbd.miller.32878.flac16 )
http://www.archive.org/details/gd73-02-09.sbd.allred.9888.sbeok.shnf - Another one of the famed "let's debut a whole new album" shows - the first performances of Row Jimmy, Loose Lucy, Here Comes Sunshine, They Love Each Other, Eyes of the World, China Doll and Wave That Flag. (Only three of these would make it onto Wake of the Flood, though, which was recorded in August '73.)
The Weather Report Suite is tricky - the Prelude had been floating around for years, Let It Grow debuted on 9/7/73, and the whole Suite was first performed on 9/8. Another of the few Dead songs that debuted only after being recorded for an album.
http://www.archive.org/details/gd73-09-08.sbd.wulf.183.sbefail.shnf (9/8/73 also sees the debut of the short-lived Let Me Sing Your Blues Away.)
http://www.archive.org/iathreads/post-view.php?id=263203 - list of early Preludes in '72/'73.
http://www.archive.org/details/gd69-04-05.sbd.miller.18701.sbesok.shnf - the earliest Prelude tease I've found. (There's also a tease in the 8/6/71 Other One reprise.)
First performances of the remaining Mars Hotel songs (recorded in April '74) aren't too distinctive, but for the sake of completeness:
http://www.archive.org/details/gd1974-02-22.sbd.gans-latvala.samaritano.26763.shnf - first Ship of Fools & Must Have Been the Roses.
http://www.archive.org/details/gd1974-03-23.aud.connors.hughey.gems.78599.flac16 - first Scarlet Begonias & Cassidy.
http://www.archive.org/details/gd1974-05-17.sbd.miller.89972.sbeok.flac16 - first Money Money (out of three).
http://www.archive.org/details/gd74-06-23.sbd.cribbs.16780.sbeok.shnf - OK, it's a cover and not a Dead song, but worth mentioning as the only Dead performance of a song off Garcia's Compliments album - Let It Rock. This show also has the debut of Seastones!
From early '75, of course, we have innumerable rehearsals of the new songs in the studio, but to focus on the live debuts:
3/23/75 has the first of the three Blues for Allah>Stronger than Dirt performances.
8/13/75 has a rare Sage & Spirit (the only other one, I think, was on 10/31/80).
http://www.archive.org/details/gd75-06-17.bertrando.unknown.233.sbeok.shnf - first Help On The Way (no vocals)>Slipknot>Franklin's Tower, and first Crazy Fingers.
(An early Slipknot riff had also been played a few times in '74, for instance 6/20 and 7/25/74.)
The first live Music Never Stopped was on 8/13/75, but the first live one on the Archive is - http://www.archive.org/details/gd75-09-28.sbd.fink.9392.sbeok.shnf
http://www.archive.org/details/gd76-06-03.sbd.bertha-ashley.20004.sbeok.shnf - first live Wheel (five years after being recorded!), Might As Well, Lazy Lightning>Supplication
http://www.archive.org/details/gd76-06-04.sbd.cotsman.9797.sbeok.shnf - not a Dead song really, but the first of five Dead performances of Mission in the Rain.
http://www.archive.org/details/gd77-02-26.sbd.alphadog.9752.sbeok.shnf - first Terrapin Station & Estimated Prophet.
Although I should note, an early Estimated Prophet riff was being practiced as early as '74!
http://www.archive.org/details/gd77-03-18.sbd.unknown.254.sbeok.shnf - first Fire on the Mountain. (However, Mickey Hart did a studio version of this way back in '71, I believe.)
http://www.archive.org/details/gd1977-05-15.28916.sbeok.flac16 - first Passenger.
Well, that's as far as I'll go with this....
It's worth mentioning that after the hiatus they introduced slower, rearranged versions of some older songs (like They Love Each Other, Friend of the Devil, Dancin', etc), but that's a different discussion.
Another interesting list could be made of the first performances of the Dead's covers - for instance, to take a random few from Skull & Roses, the first versions of Not Fade Away, Goin' Down the Road, Me & My Uncle, and Mama Tried are rather different than what they became. I'm putting together that list as well....