Fame is the name of the game
Fame is like reputation points, or how well known your character is in a location. The higher it is, the better standing you have with the NPCs of that area. The better your fame, the more quests and options are open to you. However, fame is not a tangible statistic. There's no reputation window where you can watch your standings slowly rise towards "exalted."
Instead, FFXI uses NPCs to communicate how well known you are. These NPCs are Flaco at E-6 in Port Bastok, Namonutice in K-6 in San d'Oria, and Zabirego-Hajigo in F-10 in north Windurst Waters. Each of NPCs will tell you how well they know you using a sentence. The better the sentence sounds, the higher your rank is in the city. Rank moves on a scale of 0 to 9, 0 being beginning and 9 being maxed.
Quests are not restricted by level or allegiance, but they can be restricted by fame. Some NPCs will not speak with you until your fame reaches a level where they recognize you. Usually the higher fame quests require you to have a higher level as well, but this is not always the case. Higher fame quests do offer better rewards, however, including items that can only be obtained through questing. Some of these include maps, special exits in your mog house, and sweet equipment.
Fame is also used to partially trigger events, like upgrading the size of your mog safe. This just adds to the list of reasons you want to improve your fame, especially in your home city.
Always take quests from small Mithra girls
One of the newbie quests that you really want to get done ASAP are the "kitten quests." In each city, there is a young Mithra girl who's looking for a type of flower. Fulfill their wishes and you'll get the extremely beneficial mog house exit upgrade, which allows you to jump around cities with ease. This upgrade will give you the option to go to any other mog house exit in the city, regardless of where you entered. Bastok, Windurst, and San d'Oria all have versions of this quest, and you'll need to do it in each city to get the upgrade.
Where can I find a nice quest list?
The one part about FFXI's quests is that they're not apparent. This is lots of fun for people who like talking to all the NPCs, but sometimes you just need to get the job done.
My personal favorite resource is FFXIclopedia (if you haven't noticed by now), as they have information on pretty much everything in the game, including listings of quests. Simply refer to their huge list if you're looking for ways to improve your reputation, or gain some maps and items that may help you in your job without straining your bank account.
Well, that's all for me for today. On Thursday, I think we're going to be re-visiting Valkurm Dunes, as it should be about time to grab your sub-job and start considering going to Jeuno. Until then, safe adventures in Vana'diel!