The main thing to understand is that there's a difference between missions & quests. The missions are done in order to progress your rank in whatever nation you started in (Bastok, Windurst, or San d'Oria) and you can't skip them. In order to do the later ones, you have to do the earlier ones. There are good reasons to do these missions so I suggest keeping up with them if possible. For example, once you reach rank 6 you'll be granted an airship pass to ride the airships which is convenient.
As for quests, they usually depend on fame in order to start the quest. Each quest will require a certain level of fame in a certain region, and you increase your fame by completing other quests in that same region. There are some quests that require you to do other quests beforehand, if they two quests are related in terms of story, but generally fame is the only requirement to start most quests.
The easiest way to increase your fame is by finding a nice and easy repeatable quest that you can do a bunch of times to quickly boost your fame. If you're looking for fame in a specific area, we can suggest a repeatable quest for you to do.
Edited, Apr 12th 2012 11:37am by Pergatory
____________________________Pergatory (Asura, Kupo LS) WHM99 ~ SMN99 ~ DRK99 "If you want to know me, look inside your heart." -Lao Tzu