Exploring War Like Worlds.

Checkout this lodestone article with map’s and instructions.

From Limsa Lominsa to Ul’dah
First, make your way to the ferry docks located on the lower decks of Central Limsa Lominsa (just beyond Hawkers’ Alley). Speak with the fellow at the counter to receive your free boarding ticket, and then proceed through the gate. You will automatically board the ferry before it leaves port.
In approximately ten minutes (earth time), the ferry will arrive in Vesper Bay.

I think you can not get through the dock until your at a certain point in the city quest. Perhaps to stop yourself getting killed. I think also a quest requires you to go to vesper bay.

You can travel without having done a quest. You just need to run up to the gates until you get an icon like for a gathering node or a door. You will also auto get on the boat if your about to board the boat when the boat starts to leave.

Step 2: East on the Sunway.
Once you have arrived in western Thanalan and disembarked the ferry, exit the dock area and follow the Royal Allagan Sunway east to central Thanalan. If you are feeling adventurous, take a slight detour and attune with the aetheryte located in Camp Horizon (13. 29). You won’t regret it!

Step 3: South on the Starway.
Midway through central Thanalan, you will come to a road leading south. This is the Royal Allagan Starway, and it can be taken all the way to the gates of Ul’dah. On the way, do not forget to attune with the aetheryte at Camp Black Brush (27. 25).