Got it down to turn 1 on the draw if we allow Mons's Goblin Waiters!
Turn 1 - infinite damage on the draw (16 cards used)
Turn 0:
Gemstone Caverns, exiling Mons's Goblin Raiders for irony (1 card "used")
Turn 1:
City of Traitors, tap both lands for 2R (2R, 2 cards)
Play Mons's Goblin Waiters for R, sacrifice both lands for R (2R, 3 cards)
Play Skirk Prospector for R, sacrifice Mons's Goblin Waiters for R (2R, 4 cards)
Play Mogg War Marshal for 1R, sacrifice it and both tokens for RRR (1RRR, 5 cards)
Play Mogg War Marshal for 1R, sacrifice it and both tokens for RRR (RRRRR, 6 cards)
Play Goblin Ringleader for RRRR, revealing Mogg War Marshal, Goblin Lackey, Goblin Bushwhacker, and Grenzo, Dungeon Warden, then sacrifice it for R (RR, 7 cards)
Play Mogg War Marshal for RR, sacrifice it and both tokens for RRR (RRR, 8 cards)
Play Goblin Lackey for R (RR, 9 cards)
Play Goblin Bushwhacker, Kicked, for RR (10 cards)
Attack with Goblin Lackey for 2 damage, put Grenzo, Dungeon Warden into play (11 cards)
Sacrifice Goblin Lackey and Goblin Bushwhacker for RR (RR, 11 cards)
Pay RR to Grenzo, revealing Siege-Gang Commander, put it into play, sacrifice it and all three tokens for RRRR (RRRR, 12 cards)
Pay RR to Grenzo, revealing Siege-Gang Commander, put it into play, sacrifice it and all three tokens for RRRR (RRRRRR, 13 cards)
Pay RR to Grenzo, revealing Goblin Sledder, put it into play, sacrifice it to give Grenzo +1/+1 (RRRR, 14 cards)
Pay RR to Grenzo, revealing Kiki-Jiki, Mirror Breaker, put it into play (RR, 15 cards)
Pay RR to Grenzo, revealing Lightning Crafter, put it into play, but hold the ETB trigger on the stack... (16 cards)
Tap Kiki-Jiki, targeting Lightning Crafter to create a token, champion Kiki-Jiki with the token, tap the token to deal your opponent 3 damage, then sacrifice the token for R, returning Kiki-Jiki to the battlefield untapped
Repeat the step above infinite times, dealing your opponent infinite damage!
Without Mons's, the best is turn two:
Turn 2 (pre-combat!) - infinite damage on the play (12 cards used)
Turn 1:
Geothermal Crevice tapped (1 card used)
Turn 2:
Tap and sacrifice Geothermal Crevice, play Frogtosser Banneret for BG (2 cards)
Mountain, play Akki Rockspeaker for R, gaining R (R, 4 cards)
Play Skirk Prospector for R (5 cards)
Sacrifice Akki Rockspeaker to gain R, play Mogg War Marshal for R (6 cards)
Sacrifice Mogg War Marshal and both tokens for RRR (RRR, 6 cards)
Play Mogg War Marshal for R (RR, 7 cards)
Sacrifice Mogg War Marshal and both tokens for RRR (RRRRR, 7 cards)
Play Goblin Ringleader for RRR, revealing Mogg War Marshal x2, Kiki-Jiki Mirror Breaker, Lightning Crafter (RR, 8 cards)
Play Mogg War Marshal, Mogg War Marshal for RR (10 cards)
Sacrifice both Mogg War Marshals and all four tokens for RRRRRR (RRRRRR, 10 cards)
Sacrifice Goblin Ringleader for R (RRRRRRR, 10 cards)
Play Kiki-Jiki, Mirror Breaker for RRRR (RRR, 11 cards)
Play Lightning Crafter for RRR, but hold the ETB trigger on the stack (12 cards)
Tap Kiki-Jiki targeting Lightning Crafter to make a token, champion Kiki-Jiki with the token. Tap the token to deal 3 damage to your opponent, then sacrifice the token to Skirk Prospector for R, returning Kiki-Jiki to the battlefield (untapped)
Repeat the above step infinite times, dealing infinite damage to your opponent!
(With the original Lightning Crafter trigger still on the stack)
As well as the most efficient kill:
Turn 2 - infinite damage on the play (6 cards used)
Turn 1:
Mountain, Goblin Lackey (2 cards used)
Turn 2:
Tap Mountain for R, play Skirk Prospector (3 cards)
Attack with Goblin Lackey for 1 damage, put Kiki-Jiki, Mirror Breaker into play (4 cards)
Play Gaea's Cradle, Tap for GGG (GGG floating, 5 cards)
Sacrifice Goblin Lackey to Skirk Prospector for R (GGGR, 5 cards)
Play Lightning Crafter for GGGR, but hold the ETB trigger on the stack (6 cards)
Tap Kiki-Jiki targeting Lightning Crafter to make a token, champion Kiki-Jiki with the token. Tap the token to deal 3 damage to your opponent, then sacrifice the token to Skirk Prospector for R, returning Kiki-Jiki to the battlefield (untapped)
Repeat the above step infinite times, dealing infinite damage to your opponent!
(With the original Lightning Crafter trigger still on the stack)
Optimizing for card efficiency another way:
Turn 2 - infinite damage on the play (5 cards from hand)
I'm proud to have broken Goatnapper, although the consequences may be tragic - after having proven that it facilitates a practically unbeatable turn two infinite combo deck, I am sure it will shortly be banned from Legacy, and we will all be poorer for the loss.
.
Turn 1:
Mountain, Goblin Lackey (2 cards played)
Turn 2:
Attack with Goblin Lackey for 1 damage, put Grenzo, Dungeon Warden into play (3 cards)
Tap Mountain for R, play Skirk Prospector (4 cards)
Play Gaea's Cradle, Tap for GGG (GGG floating, 5 cards)
Pay GG to Grenzo, revealing Siege-Gang Commander, put it into play, get three Goblin tokens
Sacrifice Siege-Gang Commander and all three tokens to Skirk Prospector for RRRR (GRRRR, 5 cards)
Sacrifice Goblin Lackey to Skirk Prospector for R (GRRRRR, 5 cards)
Pay GR to Grenzo, revealing Amoeboid Changeling, put it into play
Pay RR to Grenzo, revealing Goatnapper, put it into play targeting Amoeboid Changeling (a Goat, as well as a Goblin), untapping it, gaining control of it, and giving it haste
Pay RR to Grenzo, revealing Kiki-Jiki, Mirror Breaker, put it into play
Tap Amoeboid Changeling (haste from Goatnapper) to give Kiki-Jiki all creature types until end of turn
Tap Kiki-Jiki, targeting Goatnapper, creating a token with haste. Target Kiki-Jiki with the Goatnapper token trigger (Kiki-Jiki is currently a Goat as well as a Goblin), untap and gain control of Kiki-Jiki (as well as giving it more haste)
Repeat the above step infinite times for infinite Goatnapper tokens, leaving Kiki-Jiki untapped
Sacrifice your infinite Goatnapper tokens for infinite R from Skirk Prospector (Infinite R, 5 cards)
Pay RR to Grenzo, revealing Goblin Sledder, put it into play (Infinite R, 5 cards)
Sacrifice Skirk Prospector and Goblin Sledder, giving Grenzo +2/+2 until end of turn
Pay RR to Grenzo, revealing Goblin Dynamo, put it into play (Infinite R, 5 cards)
Tap Kiki-Jiki, targeting Goblin Dynamo, creating a token copy with haste
Tap and sacrifice the Goblin Dynamo token with X = infinite, dealing infinite damage to your opponent!