Mounts and Pets
Alright, lets talk about one of the most awesome things about WoW. Mounts and pets. Every race has their own special mounts, and you can buy the mounts of other races (within your faction or of neutral factions) once your reputation with them is high enough. We'll cover reputations after this.
Mount types are as follows
Alliance
Humans - Horses
Dwarves - Rams
Gnomes - Mechanostriders
Night Elves - Nightsabers
Draenei - Elekks
Worgen - Mountain Horse
(Note: Worgen originally didn't have their own mounts. Instead they had an ability called "Running Wild" where they'd just run on all fours. However, this was pretty unfair to people that faction changed and lost their previous mounts. Whenever you change factions, all your existing mounts get changed to the mounts of the other faction. So if you have an orc and have all the wolf mounts, then decide to switch your orc to a human, all your wolves get changed to horses. But because worgen had no mount, when you switched, your amount of mounts would decrease. So Blizzard gave them horses without saddles to fix this issue.)
Horde
Orcs - Wolves
Trolls - Raptors
Tauren - Kodos
Forsaken - Skeletal Horses
Blood Elves - Hawkstriders
Goblins - Trikes
Neutral
Pandaren - Dragon Turtles
Alliance flying mounts are Gryphons
Horde flying mounts are windriders.
There are also dragon mounts, bears, mammoths, mechanical mounts, hippogryphs and more.
At lv20 you can get your first ground mount. You'll have to pay for training and pay for the mount itself, but its not TOO expensive. At lv40 you can upgrade your riding ability and buy a faster ground mount.
At lv60 you can upgrade again and get your first flying mount. You can also buy the Flight Masters License which allows you to fly in Azeroth. At lv68 you can buy Cold Weather Flying to fly on the continent of Northrend. At 70 you can upgrade to a faster flying mount.
Mounts can get pretty expensive, which is why you need jobs to make gold. I had like 60 mounts the last time I played and some of them I spent a LOT of gold on. Just averaging it out by checking mount prices, I probably spent about 6,378g on mounts. And thats nothing compared to what some people will spend. The Travelers Tundra Mammoth costs 20,000g (16,000 with exalted rep).
Companion Pets
Unlike hunter pets, these small vanity pets are for looks mostly. You can find them all over the place. You can get them as quest rewards, drops in instances, special holiday events, you can even find and capture them out in the wild.
Vanity pets follow you around and don't really do much of anything. However, you can use them to battle other peoples pets in a Pokemon style battle. I don't think theres any limit on how many you can have (though you can only have one out and active at a time). I had 76 of them.
Pets come in all different types. Cats, dogs, bugs, dragon whelps, snakes, birds, etc.