Question in title. Assuming a fresh run, so no twink gear.
First off nice name 
For me, i have always had an easy time in normal/veteran using night blade and duel wield, and or demo with canister bomb and fire strike. It all really depends on what kind of play style you are looking for. 
Easiest run through normal I had was a demo using Blackwater cocktail with it’s transmuter, mainly to proc Fissure devotion
Fire Strike 12/12 and then rush to get Brimstone 12/12.
You will have a lot of health very quickly meaning you will be able to withstand a lot of damage and not have to give much concern to large damage spikes.
Then once you reach Brimstone and max it out you will just watch the mobs in Act II explode.
Love my Fire Striker 
Syn
I haven’t tried a demo build in a while, so I think I’ll go with that. Thanks for the input, guys.
Good luck man
Vitality Conjurer
Any spellcaster. Slap a Searing Ember into a +% fire damage offhand and just destroy everything in your path with Fireblast all the way to level 20+, if you stack +%fire/elemental damage mods. Upgrade to Flintcore Bolts as required, slap a devotion in there when you can.
Summoners are particularly powerful at these levels because you end up combining absurd minion levels (max them out ASAP) with a nuke that can potentially kill mobs in 2 or so hits. And you can add devotion procs to all of them.
Anything with a shaman is pretty easy. If you rush briarthorn to max level he will be hitting with an aoe attack for several hundred damage by the halfway point of act 1, then you can always respec the skill points later. I found my vit conjurer was pretty easy to level. I used briarthorn until Wendigo and storm totem were leveled a bit and now she does a lot of damage and heals for a ton. There are a lot of good devotion synergies for a vit caster. As mentioned, a summoner is also fairly easy and you can shop a lot of pet damage gear from vendors.
I haven’t tried it myself, but a Shaman with Primal strike and a decent 2hander + the Kraken devotion apparently destroys everything until Elite and is also a good endgame build at the same time.
Normally I’d agree with you but he said no twink gear, which means he may not get his first non-twinked Searing Ember early enough nor get enough Searing Embers for Flintcore Bolts early enough. If he is willing to twink components then I would add (Noxious) Poison Bomb to the mix as well. It helps against fire-resistant enemies, stacks DoT with Fireblast and the Noxious version even reduces resistances. Ice Spike also works but not Greater Ice Spike because Chilled Steel is better used for Haunted Steel. I find Lightning Nova to be too energy-hungry.
Any class except perhaps Soldier can get away with pumping PURE mastery points for 17 levels just by using Fireblast + Poison Bomb starting from level 7 and a 2H rifle before that. The spirit bonus from the mastery points alone boosts damage significantly. Then the character can use a strong Tier 9 mastery skill like Blade Spirit - which, BTW was recently buffed to be undisputedly one of the best leveling skills starting from level 20-ish.
Searing Embers are extremely common (you can even buy them off merchants) and a low level off-hand with +%fire damage only costs 500-1k iron bits (depending on number/quality of extra affixes). I always manage to get one around level 7-10 on my characters, which is why I never bother actually farming for them anymore and just keep a low level pre-socketed +%fire damage offhand ready in my transfer vault. You only need to do it once, and you can’t use it until you reach those levels anyway.
Ill chime in…
Maxes shaman’s swarm will faceroll the entire 1st act mostly.
Wendigo Totem is easily the most OP skill in the game. You can max it by level 12. You’ll have an easy time the whole game, not just early. The AoE damage will carry you a long way into normal.
Yeah Devouring swarm will get you through a large amount of normal/veteran really easily.
+1 for shaman. I don’t like using swarm but it works well. PS is busted early on and shows no signs of slowing down till ultimate with crap gear, wendigo totem is awesome. You can play with only your mouse, just drop totem from mouse button and hold down left click for entirety of vet/elite. Very easy 
Shaman is probably the easiest with just devouring swarm. Though with the changes to Summon Familiar Occult is right up there now.
Anytime I start a fresh toon, regardless of class, without using saved or crafted gear I get Searing Embers to put on my weapon ASAP and just spam Fireblast. It’ll take you through a good portion of act 1 until you get some epic drops.
I made a mule the other day and once I got a few points in familiar it was a breeze getting through that first quest.
For early game normal, the best two skills are Devouring Swarm and then Shadow Strike by a fairly wide margin. Those two skills make up the speedrun skill route for a reason. 