(Quick note, for some of these, I'd recommend getting the Encounter Power Dragon 1 meal from the Seabreeze Café in Mesagoza. Also, skip to the end for the TLDR of the dragons themselves.)
The earliest two earliest Dragons in Violet are Goomy and Dreepy.
These two can be found near the ponds by Los Platos. Exit the town at the left side and go up the hill. Follow the path on the Southwest, and there will be a small gap you can jump over with Miraidon. Continue going up that path, and there will be a pond where Dreepy and Goomy can spawn next to it.
I included an image, which hopefully you can see:
I think the third quickest would be Applin.
Applin can be found on trees in various locations, but with a Dragon Encounter Power sandwich, I found one by headbutting a the big tree in the area South of Cortondo. However, I think headbutting requires having the dashing ability unlocked by beating the Titan Klawf, so since you have the DLC, I'd recommend getting an Applin from the Apple Fields in Kitakami. While you're there, I'd also recommend going to the Mossfell Confluence and buying a Syrupy Apple if you want to use Dipplin. Flapple and Appletun can also rarely spawn in the field near there if you're interested in that.
So yeah, Goomy, Dreepy, and maybe Applin would be your quickest options.
However, these
SUCK!!!
Applin and Dreepy in particular have atrocious learn sets, and while Goomy is a bit better...
Are those really the stats you want to be stuck with until level 40?
LUCKILY, Axew and Bagon can also be found early. You'll want to go East from the Pokemon Center in front of the Mesagoza gates (in South Province Area 1 where you battle Nemona) then South to the elevated ledge. There's another gap you can clear with Miraidon. From there, the directions are hard to describe, but you'll want to stay on the elevated part of the terrain and make your way Northeast to the big spiral mountain in South Province Area 5. I again included a picture, which hopefully you can see:
I think the biggest problem you will encounter is being able to catch and use various Pokemon. Many of the dragons you will encounter will be higher level, but if you really need to, you can always reset your game to ensure a catch and breed to get a level 1 Pokemon, with the obvious caveat being you will have to train it up. I do think this is worthwhile to do if you want to use better dragons earlier, especially if you're willing to go out of your way to get them and don't want to deal with potential obedience issues.
For some other options, there's a cave in the wall at the Southwest edge of the dark fields terrain across the river from Cortondo, where Gible (and Bagon) spawn. Here's a picture:
If you're open to using mons that aren't immediately dragons but evolve into them, Swablu can also be found near the spiral mountain, and Trapinch can be found in the Savannah biome of the Terrarium (you don't have to win the battle at the intro of the Blueberry Academy to get into the Terrarium). You will want to breed for Trapinch as it will be high level iirc. There's also Duraludon in the Polar Biome, which I think you can reach by following the border of each biome to the center and entering from there. It's a bit more out of the way but still an option.
TLDR: The earliest Dragons are Goomy, Dreepy, and possibly Applin, but since the first stages kinda suck, you might want to grab Axew/Bagon, go out a bit further for better dragons, breed for a low level dragon, and/or expand your options to allow something that evolves into a dragon but isn't one yet.
Hope I helped!
