Class Quests
Class Quests are replaced by Job Quests after level 30 for combat classes. There is no reason to be playing the base class once a job is available.
Crafting and Gathering classes instead have quests up to level 70, and then progress to The Crystalline Mean in Shadowbringers, followed by The Studium in Endwalker, and Wachumeqimeqi in Dawntrail.
Disciples of War Quests
Disciples of Magic Quests
Disciples of Hand Quests
Most Disciples of Hand (crafter) quests will require the player to turn in an item(s), which will usually be required to be high quality for higher-level quests. These items can either be crafted by the player or be obtained from vendors (NQ only) or the Market Board, which may be a convenient, though potentially expensive, option. If they choose to craft them, the player must also procure the requisite materials. Starting from Stormblood (Lv 63+), the quest turn-ins will be UNTRADABLE and require the player to craft them. However, materials (which are also quest-specific) will be provided by the quest NPCs and can be renewed if the player loses them, so they will not need to acquire the materials themselves, unlike in previous quests.
It is recommended to complete class quests as they become available due to being a good source of 
experience. In addition, completing the Lv. 65 quest unlocks Manipulation for the respective class. This important action is used in many high-level crafting rotations.
Disciples of Land Quests
Starting from Stormblood (Lv 63+), the quest turn-ins will be UNTRADABLE and require the player to gather them. The requisite items are only available during the respective quests, so the player cannot gather them preemptively.