There have been various discussions about this question, and the answer depends upon the play style:
1) Self multiplay- enchantress in game means very speedy progress, the limitation is walk speed so an assassin (with Burst of Speed) or paladin (Vigor, but doesn't get Vigor until level 18) are good choices.
2) Single-player twinked: an Envy setup works well, and another popular method is to use a 1.09 or earlier Ravenclaw for very speedy killing through normal. Getting good FRW on low-level items isn't so easy- I like Hsarus' boots and belt for that. Some people like stamina-related items for the early running, but with muled-on money (easier for level 1 characters these days) you can just prebuff a long long run with shopped stamina potions.
3) Single-player untwinked: this is the most interesting case to me. Sorc with tele is commonly suggested as a best choice here, as people remember how quick teleport is. But two things limit that- you don't get teleport until level 18, and you don't have a ton of mana like you are used to with well-equipped higher-level characters. So I prefer a Wake of Fire assassin which has an early Burst of Speed and very strong killing through normal. Another good choice is a Holy Fire paladin for a quick start. I recall at least one tourney which involved a quick trip through normal untwinked and an HF paladin got off to a very quick start for me. Both the WoF assassin and the HF paladin slow down immensely later in NM but through normal they are very quick.
For going through all three levels (norm, NM, Hell), a sorc is the clear favorite and Nightfish's methods are fabulous for that. But for just going through normal, much of the time for a sorc is spent on those first 18 levels. A fireball sorc is reasonably quick and there is a chance you can make a good Leaf staff (shopping for bases in A2 norm if you do find the runes.) With some luck, a fire sorc could definitely be faster than pally or assassin but I think not as dependable. In HC, I would prefer a WoF assassin.
In terms of accumulating socket quests, you may also want to consider rushing characters. I have many many socket quests left over from rushed characters, many from 1.07 rushes, as those come at a rate of about 7 at a time.
Edit: a few other thoughts:
muling on Khalim's Will is a great way to have a very quick start in A1. It can avoid needing to visit the all of the dungeons in A3. Of course, if you do that you'll use up the KW when you smash the orb. You can instead farm Khalim's Wills with a sorc beforehand (for example, a strong sorc in A3 Hell who hasn't smashed the orb yet can make those pretty quickly with a fixed map) if desired. Or you can mule KW off when the character is entering A3, for another level 1 character to have a speedy start.
I did a writeup for a Wake of Fire assassin,
KVSwizter which may have some useful bits in there. That character had additional limitations (no WP or TP use allowed) so time-wise, it took much longer but the build is appropriate for a quick trip.
I found an old quick HF pally, he was in the
Ten Days, Ten Tasks tourney of AJK some time ago but I don't think there was much description there. The object was to get 10 perfect gems in one day with a new character, so I made an HF pally to get through normal (with a forge perfect gem) and then farm cows to get enough flawless to cube.