Most quests are not repeatable. Repeatable quests such as Beast Tribe dailies have a special quest marker with two arrows forming a circle.

For alt classes you can level them with class Hunting Logs, Fates, Duty Roulettes, spamming the highest level dungeon available to you, PVP, Palace of the Dead, leve quests, ect.
Make sure you've unlocked the Challenge Log (quest: Rising to the Challenge in Limsa Lominsa). It gives you bonus exp and gil for meeting weekly objectives.