At level 60, there are a few optional things you can unlock.
First: Wondrous Tails. (No, that's not a typo.) In Idyllshire, speak with Khloe Aliapoh, a young Miqo'te. She hands out a journal of challenges you can complete. There are 16 per journal, but you only have to complete 9. Each one puts a random sticker in the journal; completing all 9 gets a chunk of experience and a reward, with bonus rewards for each complete line they make. You can only carry one journal at a time, it's good for up to two weeks, and the timer resets on Tuesdays. The quest explains it better than I can.
Next: Alexander. Speak to Slowfix in Idyllshire to get the first quest. Alexander is a normal raid series, consisting of three sets of four trials apiece, for a total of twelve. Aside from fun fights, it's also a good source of tomestones of poetics, used to buy endgame gear in all but the latest expansion. It'll also unlock the Normal Raid daily roulette in the Duty Finder window.
Third: The Shadows of Mhach, a set of alliance raids (24-player dungeons, like the Crystal Tower raids back at level 50). You'll have to have unlocked flying in the Sea of Clouds to access this. Speak to the Unquiet Trader in Ishgard near the airship docks to start it.
Fourth: The Warring Triad. Speak to Torsefers in the Pillars in Ishgard. Unlocks a side story containing three trials against, well, the Warring Triad. If you just hit 60, you might have to progress a bit further into the main story to access this one, you might not have met a couple of the characters involved yet.
First: Wondrous Tails. (No, that's not a typo.) In Idyllshire, speak with Khloe Aliapoh, a young Miqo'te. She hands out a journal of challenges you can complete. There are 16 per journal, but you only have to complete 9. Each one puts a random sticker in the journal; completing all 9 gets a chunk of experience and a reward, with bonus rewards for each complete line they make. You can only carry one journal at a time, it's good for up to two weeks, and the timer resets on Tuesdays. The quest explains it better than I can.
Next: Alexander. Speak to Slowfix in Idyllshire to get the first quest. Alexander is a normal raid series, consisting of three sets of four trials apiece, for a total of twelve. Aside from fun fights, it's also a good source of tomestones of poetics, used to buy endgame gear in all but the latest expansion. It'll also unlock the Normal Raid daily roulette in the Duty Finder window.
Third: The Shadows of Mhach, a set of alliance raids (24-player dungeons, like the Crystal Tower raids back at level 50). You'll have to have unlocked flying in the Sea of Clouds to access this. Speak to the Unquiet Trader in Ishgard near the airship docks to start it.
Fourth: The Warring Triad. Speak to Torsefers in the Pillars in Ishgard. Unlocks a side story containing three trials against, well, the Warring Triad. If you just hit 60, you might have to progress a bit further into the main story to access this one, you might not have met a couple of the characters involved yet.