Mounted pets are a great way to get from one location to another quickly, sometimes by flying!
General Shop NPCs sell "Wings of the Goddess", which will teleport you to the last landmark or town you visited. Additionally, Moon Gates and Mana Tunnels are scattered throughout Uladh and Iria which allow quick teleporting between popular destinations. If you need to travel between Uladh, Belvast and Iria quickly, you can use the "Continent Warp" action once every 12 in-game hours, available in the "Actions" menu (hotkey [N]), located under the third icon (Skills and Actions).
To reach a Moon Gate or Mana Tunnel, you can also use the Traveler's Guide! Enter a location you'd like to travel to, and you can choose the closest Moon Gate or Mana Tunnel that'll suit your destination! To access the Traveler's Guide, use the default hotkey "[" or click on the Mabinogi symbol icon on your in-game taskbar. This is the second icon from the left of the screen: