Welcome to Eorzea!

You're up to the part in the story where you travel outside your starting city to the other two citystates as part of the storyline, with Limsa Lominsa your first destination. You should have been given your airship pass so you are required to travel by airship as a part of this, thus head to the Airship Dock which can be found in the Hustings Strip, and then just follow it from there! (airship travel between the three cities and travel on the ferry between Limsa and Vesper Bay is then unlocked after that)

Also your level is fine - the point in the game you're up to is around level 15-17 so you're not too overlevelled (and as you have your very first instance dungeon coming up being a higher level can help out in that regard). Either way, good luck!