Okay, here is my guide to making the Ultimate Level 1 Twink. With this twink I've dueled and beat many people 5-8 levels higher than me - my best achievements so far are a level 14 warrior and level 13 pally, and an 18 lock, but the lock was without pet. That's with no buffs, potions or anything - just my gear.
Before I begin... if you want to flame twinks or say "what's the point of this?" then just move along and don't bother posting. Some people have fun gearing low level characters to see how they can compare against higher level characters. So scram. Okay, now that's out of the way...
CLASS
IMO the ultimate Level 1 Twink class is a rogue. I've dueled a warrior as decked out as I am, and managed to beat him. If you don't want a rogue though, then warriors are nice, and I would think pallies would be pretty good too. But for the purposes of this guide, I'll be enchanting as if for a rogue. If you're playing something else, choose your enchants accordingly.
Note: If you choose a Rogue, you can actually learn Stealth at Level 1. Just go to the class trainer in your starter area and they will teach you the ability at Level 1 (it's a common misconception that you can't get it until Level 2 - even funnier when you kick a higher level's butt and then they claim you're a "hacker" because you have stealth! lol)
I chose: Rogue
RACE
If you're making your twink Alliance-side, I'd go with either a human or a night elf. Humans get a +5 specialization to swords (which you'll be using if you get the sword I recommend) whereas night elves have a +dodge racial which is definitely very nice. I made my toon a human, but a nelf may be better (too lazy to look up how worthwhile the sword specialization is - I didn't want to make a nelf anyway). That said, if you're not making a rogue, I would go draenei. Gift of the Naaru is imba, I wish they had draenei rogues! Horde-side, well, I'm not familiar with their racials, but pick whatever you think is most useful.
I chose: Human
BEFORE YOU BEGIN...
The trickiest thing about a Level 1 twink is making sure you don't accidently level it up! You get a huge amount of XP just for discovering parts of the map, or accidently killing an animal, etc. so you have to be careful. Because I chose a human, I started in Nortshire, ran down to Goldshire, and then over to Stormwind to get sword training. Then I levelled up my weapon skills on critters, and my defense skill by letting mobs hit me (and NOT fighting back - you don't want XP). Try not to run off the road anywhere and discover places because it'll give you XP.
I also picked up first aid while I was in Stormwind (or any capital city). Send your twink and a heap of linen cloth and wool cloth, and they'll be able to skill up their bandaging. Note however that when you have such a huge health pool (over 600 HP) that the bandages you can make are almost worthless to you.
GEAR
Head: Red Defias Mask, Noble's Monocle
This was the most annoying item to get, because it's BoP. It has no stats, but you need it so you can apply a head enchant!
Red Defias Mask - Item - World of Warcraft
Red Defias Mask is only equippable by rogues. It drops from any defias npcs, about a 1 in 50 drop. I got three of my friends to go out to westfall (one was a priest, and one a warlock). Then they went out to an area packed with defias, summoned me, and I let the mobs kill me (do NOT let your friends kill any mobs while you're still alive). Then I ran back to the spot where I rez, but DON'T REZ. Your friends kill all the mobs until a Red Defias Mask drops. At that point you rez (make sure they've stopped killing stuff), loot the mask, and then hearth out. You'll get minimal XP for discovering one place on the map (Moonbrook, for example) but as long as you've only been to your starter town and city, you won't level off it. Don't worry, this was the hardest bit.
Note: If you have a twink that isn't a rogue, you can get Noble's Monocle. It's a rare drop from the fishing daily reward, sometimes you might see it on AH, but it's usually a prized item for bank alts. Probably harder to get than a Red Defias Mask, but unless you're a rogue it's your only option at the moment.
Note #2: If you want to put a libram on your head piece (best enchant for head - you can get 8 agility or 100 health) then you'll need a head piece that isn't BoP (for example, Noble's Monocle, as mentioned above). You won't be able to put a libram on the Red Defias Mask, because the arcanum (which you have to get on another toon) is BoP, and so is the mask.
Noble's Monocle - Item - World of Warcraft
Neck: Voice Amiplification Modulator
You can get this if you have an engineer main (or friend). Reduces the length of interrupt or silence effects on user. I didn't get it, because I'd already hassled my engi friend enough getting him to come to Westfall with me. Didn't think it was worth it for a rogue. If you just want something gimmicky for neck slot, get Soap On A Rope (more on this later).
Schematic: Voice Amplification Modulator - Item - World of Warcraft
Back: Wispy Cloak, Linen Cloak
Wispy Cloak - Item - World of Warcraft
Drops from Lost Soul, a "rare" spawn in Tirisfal Glades. I say "rare" in quotes because it's not rare at all - in fact it respawns immediately after you kill it. The cloak drops pretty much every time. Not BoP, so you can just take your main and get it.
Note: Alternatively a tailor can make you a Linen Cloak, which has the same stats. But the Wispy Cloak is cooler, and it's not like it's hard to get.
Linen Cloak - Spell - World of Warcraft
Chest: Simple Black Dress
You need this dress because it's considered to be ilevel 35 or higher, meaning you can put +150 health on it (more on this later). So you need to get a tailor to make this for you. It has no stats on it, but being able to put a high level enchant on it makes the missing stats worthwhile.
WARNING: Tuxedo Jacket is no longer considered an ilevel 35+ item. It had it's stam removed, can no longer have +150 HP chest put on it, it's it's useless for twinks now.
Simple Black Dress - Spell - World of Warcraft
Wrists: Cracked Leather Bracers
Can buy these from a vendor. Simple!
Cracked Leather Bracers - Item - World of Warcraft
Hands: Worn Leather Gloves
These have 1 armor more than Cracked Leather Gloves (vendor item from same place as bracers) but are about a 1% drop from kobolds, so judge for yourself whether you could be bothered doing this. I did.
Worn Leather Gloves - Item - World of Warcraft
Note: After about 200 kills it still hadn't dropped for me, so I put out a message in General saying I'd pay 20 silver to anyone who had them, so check your bags! Someone replied right away. Much easier than farming kobolds (although it was good to farm up lots of linen, low level pots, candles, scrolls, apples, etc.)
Waist: Squealer's Belt
This drops from Squealer Thornmantle in Mulgore. A bit of a trek for Alliance players, but you can just go get it on your main, so it's easy enough. Drops pretty much every time. Has slightly more armor than the cracked leather belt from vendor (1, I think) so it's up to you. Again, I did.
Squealer's Belt - Item - World of Warcraft
Legs: Tuxedo Pants
Like the chest, this is above ilevel 35, and it also has nice armor on it anyway! Get a tailor to make it for you.
Tuxedo Pants - Spell - World of Warcraft
Feet: Cracked Leather Boots
Vendor item, another easy one.
Cracked Leather Boots - Item - World of Warcraft
Weapon: Keen Machete, Simple Dagger
This drops off a mob in Teldrassil. Note that Simple Dagger also has the same stats as this sword, but Keen Machete looks waaaaay cooler (and that's what it's all about, right?) so all you have to do is go kill a rare mob in Teldrassil to get it. It's BoE, so you can get it on your main, just don't equip it! Check out how cool it looks (especially with crusader!) It's worth noting, though, that Simple Dagger is not BoP, thus you can trade it to another toon if you accidently level to Level 2, which is nice.
Keen Machete - Item - World of Warcraft
Simple Dagger - Item - World of Warcraft
Offhand: Furbolg Medicine Pouch (didn't bother with this, up to you though)
Supposedly you can get a Furbolg Medicine Pouch by having someone in your group grind rep while you're dead.. there's no way I cbf testing this. Just including it for the sake of completion.
Furbolg Medicine Pouch - Item - World of Warcraft
Trinket: Insignia of the Horde/Alliance (didn't bother with this, up to you though)
You can get an Insignia if you world PvP enough:
Insignia of the Alliance - Item - World of Warcraft
Quite a pain, though. Not so bad now that diminishing returns is gone, I guess. Level 1s have no spirit rez sickness.
Ranged
If you want ranged weapons, here's some info. I personally didn't bother, because to train bows I need to go to Darnassus, and I didn't want to risk hitting level 2. Best bow is Cadets Bow (drops from low level mobs, can also be fished in low level areas), and best arrows are Feathered Arrows (which come from a higher level quest, but they can sometimes be found on the AH, or you can do the quest on another toon and then trade them across). You can also attach a scope to your bow.
Cadet's Bow - Item - World of Warcraft
Feathered Arrow - Item - World of Warcraft
Novelty Items
These serve no purpose, but you can get a neck piece (I got Soap On A Rope from Griftah in Shatt) if you didn't get the Voice Amp Module, and rings (I got 2x Mood Ring from Arred in Exodar) and of course a tabard and shirt. You can't enchant any of those slots, and they have no stats, but they're fun if you want to deck out your twink. Sadly, there's no shoulders available at level 1.
ENCHANTS
Head: Libram of Voracity (8 agi) or Libram of Constitution (100 HP)
Libram of Voracity - Quest - World of Warcraft
Libram of Constitution - Quest - World of Warcraft
Cloak: Dodge (12 dodge), Lesser Agility (3 agi) or Superior Defense (70 armor)
Formula: Enchant Cloak - Dodge - Item - World of Warcraft
Formula: Enchant Cloak - Lesser Agility - Item - World of Warcraft
Formula: Enchant Cloak - Superior Defense - Item - World of Warcraft
Chest: Exceptional Health (+150 HP)
Enchant Chest - Exceptional Health - Spell - World of Warcraft
Bracers: Greater Stamina (7 sta)
Formula: Enchant Bracer - Greater Stamina - Item - World of Warcraft
Gloves: Superior Agility (15 agi)
Enchant Gloves - Superior Agility - Spell - World of Warcraft
Pants: Clefthide Leg Armor (30 sta, 10 agi) or Nethercleft Leg Armor (40 sta, 12 agi)
Clefthide Leg Armor - Spell - World of Warcraft
Nethercleft Leg Armor - Spell - World of Warcraft (hefty mats!)
Boots: Greater Stam (7 stam) or Greater Agility (7 agi)
Enchant Boots - Greater Stamina - Spell - World of Warcraft
Enchant Boots - Greater Agility - Spell - World of Warcraft
Weapon: Crusader, Lifestealing, or Fiery (I recommmend Crusader)
Enchant Weapon - Crusader - Spell - World of Warcraft (the best, IMO)
Enchant Weapon - Fiery Weapon - Spell - World of Warcraft
Enchant Weapon - Lifestealing - Spell - World of Warcraft