I practically have no mounts. What are some quick and easy mounts I can pick up?
Start buying your racial mounts.
The Twilight and Black Drakes (Already suggest before my edit)
Bronze Drake (Already suggest before my edit)
You can spend some time on Isle of Giants and kill all the NPCs there. They have a chance to drop an egg which can give you one of the three Primal Raptors.
Drake of the North Wind and Vitreous Stone Drake are two mounts you can farm on normal mode. So you can do the Dungeons over and over again till you get the mount or you get tired of it.
You still need to get the Warfront mounts like Blackpaw. You can only loot them once per cycle, but the Warfront mounts have a pretty decent drop chance compared to other mounts.
This list above should help get you started. You have a huge list of dungeon/raid mounts you can go after. There are raids like Throne of Thunder that can drop two mounts and you will get about 2-3k gold. You can use the gold to get from Legacy Raids to help you get some of the Vendor mounts you need. While your in Dungeons make sure you put on a Tabard so you can gain rep from one of the other Ally races so you can eventually get their mounts as well.
Build the stables in your garrison so you can tame 4 mounts and also there’s a garrison mission that awards a mount.
You don’t have Bronze Drake, go solo the Culling of Strat dungeon in Caverns of Time and after you go through the bookshelf (you’ll see) and kill all the zombies down that street you can either go right and kill last boss or go left and kill an optional boss who drops the mount.
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You can get the 10 man/25 man drakes from um… that raid under Wyrmrest Temple. I’m sorry I can’t remember the name right now… I need caffeine. Badly.
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Obsidian Sanctum 
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Also I’d like to suggest the website https://simplearmory.com/#/ and if you like collecting you should checkout https://www.dataforazeroth.com and https://www.curseforge.com/wow/addons/all-the-things
Also if you’re willing to invest some time on getting a neat mount, I’ll suggest going for your class hall mount, those looks really cool and unique. You need to do your legion class hall missions and the whole questline in Broken Shore (for the achievement “Breaching the Tomb”).
Priest’s class hall mount:
https://wow.zamimg.com/uploads/screenshots/normal/611172.jpg