By flying carpet, there are personal ones that you can quest for at level 45 and there are "public transportation" ones that are in specific locations and you can take at any level.
This one is Aly's personal one that she quested for.
Griffons: These are "public transportation". Some anyone can take at any level. In other areas you have to do a quest to get permission to use them.
Horses: one of the types of personal mounts you can purchase. I'll leave it to others (hint, hint) to provide screenshots of ones that I don't have. Llens horse:
This next one I don't have anymore, when Aly got her flying carpet, I sold her horse and used the money to get Llen a horse.