The harbor is your first location, from where you can run off to different neighborhoods. You can find the Aetheryte square quickly, the other parts are already laid out gradually. Basic insight: Tulliollal is built on a hill, so it has an even larger vertical footprint than, for example, Old Sharlayan in Endwalker. While the harbor is down, the royal palace is all the way up, and you can wander into different nooks and crannies.
Source: N4G PC First Impressions: Final Fantasy XIV: Dawntrail (SECTOR)