Quests are primarily to level at least one Class -> Job throughout the entire story experience. You may be able to squeeze more in, but you won't be able to rely on them.
Keep in mind that the game is set up such that classes that are below your highest level class will be able to get experience more readily in order to catch up to the highest class.
Experience can be received through a number of methods:
1. Grinding Monsters - The old fashioned way, this is the least effective.
2. Hunting Logs - Each class has their own individual hunting log. Completion of an entry rewards experience. Completion of the page (one page contains 10 entries) also rewards additional experience. No new hunt logs were added in 3.0 however there are Hunt Marks which are similar and reset every day, 15 target groups that give 20-25k experience each per set of 1-3 enemies.
3. Levequests - Levequests are repeatable quests that can be received by a Levemete at many of the towns and outposts scattered about. You want to save these for crafting.
To unlock Levequests at the town/outpost, you must complete the quest related to that Levemete (usually in the form of a Levequest). To initiate a Levequest, first you must accept the quest from the Levequest NPC. From there, proceed to the area outlined in the Levequest. Once in the area, open your Quest Log, click on the Leve and press Activate. This will start the Levequest.
Limitations - You may only have 16 Levequests at any given time. Furthermore, you can only take a Levequest if you have an allowance greater than zero. Three allowances are given every 12 hours.
4. Guildhests - Guildhests are a party based instance designed to teach mechanics of the game. These instances usually have some underlying gimmick or idea behind it. They are accessed through the Duty Finder. Note that the large allotment of experience is only provided once.
5. Dungeons - Dungeons are unlocked as the player proceeds through the Main Story Quest, although some dungeons may be accessed just by having the right level and talking to the right NPC. Dungeons are a very fast and effective way of leveling.
Once unlocked, the dungeons may be accessed through the Duty Finder.
6. FATE (Full Active Time Event) - Periodically, an event in a blue circle will appear on the zone map. This is the boundary of a FATE. Any person that enters this boundary will be automatically entered into the FATE. These events can be joined at any time and are objective based. The experience reward is based on how much the player participated in the objective. FATEs are one of the most effective ways of leveling after you have finished your roulette.
7. Duty Roulette - Duty Roulette Low Level and Guildhest will also reward an experience bonus that scales with your class's level. This bonus can be acquired once daily and will become available again after JST Midnight. Duty Roulette only requires at least two duties of the category available for use. This is the single most effective way of leveling once per day.