Airship you need Rank 5. Depending on your job (if you're a RDM or WHM it's easier) and the people you know, you will get this ~lvl40. There isn't a minimum level or anything as it is possible to get a lvl1 mule to rank5, but I'd say it's about when most people get rank5.
The exception to this airship pass is the Kazham airship which you can get at level 25 after turning in 3 keys to a guard in Jeuno. Kazham is where you will be leveling from 24/25-31 or so. (quest is here:
http://ffxi.allakhazam.com/db/quests.html?realm=&fquest=215)
So yes, you'll be riding that chocobo all over for awhile.
Now, Signet.
Signet gets you Conquest Points when you kill monsters that give you exp (too weaks do not count), it doesn't have anything to do with fame. Also, signet is the way certain nations control regions each week for Conquest. Killing monsters with it on gains you more influence for your home nation in the region you're in, which lets you do outpost supply quests in the regions your nation controls. Your nation's place (and the regions they control) in Conquest also determines what shops sell in your home nation. Conquest points can be used to get gear or other items. You can spend your points at the guards in your nation that cast signet on you, but not at the guards who are located at outposts.
There's more to do with Conquest and region control, but I won't get into that here.
Ranks and Missions...
Your rank bar does go up with crystals you turn in, but it will not allow you to gain a new rank. That can only be done through completing missions the gate guards give to you. What the crystals are good for, though, is getting access to missions if your rank bar is not full enough. Delivering packages to the outposts will only get you Conquest Points, not raise your rank.
A list of all the missions is here:
http://ffxi.allakhazam.com/db/quests.html?fqtype=Mission
Keep in mind that not all of these are required, so if theres a tough one that's skippable you can just turn in more crystals to get the next one.
The only thing that raises your fame in nations is doing quests.
edit - forgot one thing, the higher rank you are, the longer signet will last on your character without having to be recasted. This is also effected by Conquest standing (if your nation is last place, it lasts longer...if first, it lasts the normal amount).