Every city and town is busting with quests to complete; find them by talking to the NPCs. While many boil down to elaborate fetch quests, others tell the stories of the characters or can be quite entertaining. While you won't get rich doing quests, they can fund the early part of your career, and provide a number of essential spells and items as quest rewards, many of which can be sold if you can't use them yourself.
The most important part about doing quests isn't the quests themselves, but the "Fame" they bring. Fame is a hidden stat that is tracked in each town and in the world as a whole, and every time you complete a quest somewhere, your fame goes up. The higher your fame, the better quests you'll be able to do, some that increase the size of both your inventory and your house's Mog Safe. Additionally, any would-be Summoners will want to build Fame early, as they won't be able to fill out their menageries of summonable Avatars without it.
Missions are longer, more elaborate quests performed to advance your national rank or the global story. While your rank within your nation doesn't mean much, many players may look at it as an indication of your competence, so it's best not to get too far behind on these missions. Attaining a higher rank not only rewards you with gil, but also unlocks more gear that can be bought with Conquest Points. But, really, the more important benefit is that you'll be able to ride the airship once you reach rank 5, allowing rapid transport between Jeuno and the three nations.
Each nations offers 10 rank quests, each of which contains multiple missions. Completing expansion missions opens up new areas you'll want much later in the game, but it doesn't hurt to start now - Chains of Promathia missions can be started around level 30, while Rise of the Zilart missions can be started once you get Rank 6 around level 50.