I know that there has been a lot of discussion about what does or does not unlock this achievement for most people. My list of what I've got is below, but tell me in the comments if you had to get anything else:
Camel (You start with one, summon him with
Horse (Easily found mount, most enemy patrols ride them)
Horse/Ox drawn cart (Not sure if both animals count as seperate, but you can ride an ox drawn cart in the quest 'New Kid in Town' found in Letopolis.)
Chariot (You spawn in one of these after unlocking the hippodrome races. To do that you must complete the quest 'Old Times')
Canoe (Plenty of these in the rivers around Egypt)
Sail Boats (Plenty of these in the larger lakes around Egypt, such as Lake Mareotis)
Not sure if there are any more, or if riding/sailing on different variants counts as a separate vehicle. Mine unlocked after I entered the chariot races for the first time.
EDIT: A Horse drawn cart and Ox drawn cart are counted as separate vehicles. Thanks from Tricka for the comment!
Let's keep cheevin' together!