There are like 4 versions of classic. Vanilla era, anniversary, Cata and SOD.
I assume you mean Cataclysm since level 20 mounts became a thing in Wrath.
It's level 40 and 60 for vanilla, 30 for tbc and then 20 for Wrath.
You need to train riding skill to use a mount. You can buy a mount from a vendor next to the guy who does the training.
So it took me probably a total of 2 hours or more of googling and just wondering around the city, but yesterday I finally found it in Stormwind.
I bought the training at the griffon trainer, not where you buy the mounts. That's where a lot of the confusion was. I kept looking all around the trading district figuring I'd be around there, but no.
Also yeah I'm playing cata. I didn't know you could change it or whatever. I just knew there was classic and retail so I chose classic, even though that's still way beyond since I last played which was Burning Crusade.