No airships. You'll need to walk if you want to get to other cities.
Go north of Ul'dah until you reach the first camp (I think it's called Black Brush). From there you can choose the main road to go either west or east.
From Ul'dah to Gridania, take the east road and STICK to it. It will eventually turn north. Just keep following it and you will arrive in Gridania.
From Ul'dah to Limsa Lominsa, take the west road and STICK to it. You will eventually come to a Ferry Dock where you can take the boat to LL.
Once you get to those cities, you can touch the Aetheryte crystals so you can teleport directly to those cities later.