Should I do everything on one character?

[Anova]

Although it is a rough estimate, you will have 5 to 10 quests per level bracket (10-14, 15-19, etc) in areas around the starting cities. While they are useful, they do not provide enough experience to level all of the classes you want. Likely, if you are gaining 90% of your exp from quests as you prefer, you will max your pugilist/monk (the job "promotion" for each class shares the exp with the base class and the class and job will level up together) and just almost max Lancer/Dragoon.

I personally did hunting logs and FATEs until level 15. After that, I usually ran dungeons if I could find a group, and did quests while I waited. I tended to avoid FATEs when possible, but it was undoubtedly the fastest and most popular way to level. There is a huge problem with grinding after you finish the quests. Fortunately, this is a problem the development team has acknowledged and they are implementing a series of rewards to levequests and dungeons to make those just as effective leveling options as FATEs. Some details can be found on the temporary producer's letter thread if you're interested.

Unfortunately, they have not mentioned any plans to add more quests or renew them. Levequests are repeatable quests, but I find the combat leves to be the same four or five variations and would rather spend them to level crafting/gathering. Keep in mind I haven't tried all the combat levequests, so there might be a few gems that I missed. Either way though, you will probably have trouble leveling after the quests are expended. I found solace in dungeons, but I can't say the same will apply to you.