Catch up on your Grand Company quests. Do XP parties. Take up a craft and play the market. Do sidequests. Unlock Monk.

The room you're hearing about is the Inn. Everyone can rest in the Inn in any of the major cities and accumulate a 50% xp bonus over time (offline time included if you log there). There's also a journal of your quests there called "The Unending Journey" that lets you replay cutscenes and a place to store one-time-only gear like AF armor.