As for that ship that runs between Vesper Bay and Limsa Lominsa, it gets unlocked at the same time as the airships, by the quest "The Ul'dahn Envoy", "The Gridanian Envoy", or "The Lominsan Envoy" depending on where you started.
That quest also marks the end of the city specific part of the game. The remainder of the main scenario is common to all players, whichever city they began the game in, and begins with "Call of the Sea".
One tip: even if you're eager to check out the new locations you've reached before moving ahead with the story, go ahead and pick up that next main scenario quest when you get there, even if you don't plan to complete it right away. We get lots of players who take time to explore, pick up additional classes and so on, and then by the time they're ready to continue the story, they've forgotten where they left off and can't find where to pick up the next quest. So it's a good idea to pick up the next MSQ whenever you finish one, so that it's there in your quest Journal for when you want to get back to it. The next quest in the line will always begin near where the previous one was completed.