Skinning is the best combined with Leatherworking, so check out my Leatherworking Leveling Guide if you want to level Leatherworking too. Leveling your Skinning will be easy, but you will have to run around and skin hundreds of mobs.
I recommend using Zygor's 1-60 Leveling Guide if you are still leveling your character or you just started a new alt. The guide is in a small window on your main screen in-game. All the detail that you need to complete the quests are included. You’ll never have to switch back to a website, or even the in-game main map to see where you have to go or what’s next. And if you are low on gold, I recommend trying this Gold Making Guide, it can help you to make a lot of gold.
Shadowlands Skinning Trainer Location
You can learn Shadowlands Skinning from Flayer Au'khem in Oribos at the Hall of Shapes. (Coordinates: 42.6, 27.6)
Oribos is the new Shadowlands city hub for both factions. You will arrive there after you finish the Shadowlands intro quests, so you can learn the new Mining skill before you head out questing.
Leveling Shadowlands Skinning (1-150)
Don't forget to enchant your gloves with Enchant Gloves - Shadowlands Gathering! This enchant brings skinning time down from 1.5 seconds to 0.5 seconds. (works with herbalism and mining too)
You can level Shadowlands Skinning to 150 at any of these places below.
I leveled my Skinning the fastest by farming Languishing Etherwyrms and Starved Etherwyrms around the marked area. They are needed in one of the starter quests in Bastion, so this area will be pretty busy at the start of the expansion. (which can be good if no one is skinning them)
Maldraxxus - 1
Maldraxxus - 2
Kill Virulent Pests around the marked area. They are a bit spread apart, but the respawn rate is fast, you will never run out of mobs to kill.