Yeah, ill agree with the others here. Doing the main storyline quests, your class quests, and any unlock quests you come across will take care of 95% of your leveling for your main class. Unlocking mounts certainly speeds up travel, and getting to where you want to be can speed up leveling so ill agree with you there. Another thing to keep an eye on to help with that is your chocobo. At level 30 youll be able to find a quest to train your chocobo mount to fight alongside you in the open world. Your chocobo will synchronize with your character level, but it has a secondary leveling mechanic where it gets skill points you can use to specialize it toward healing, tanking or dealing damage(if you level it high enough it can master all 3).

Oh, and dont worry about the confusing map too much. You happened to start in the most confusingly-laid-out of the 3 starting cities. Which i guess makes sense. Gridania and Ul Dah are towns that have mostly stood for hundreds of years, laid out around a river and an oasis respectively. Limsa is largely made up of beached ships with rigging thrown up around them, and an active port.