The dragon fight is Home Country mission 2-3 to enable you to get Rank 3, its a lvl 25 capped fight for a maximum of 6 people. If you search for the mission in the foums there are many stratagies posted.
If you want to know about all the missions then go
here to wikipedia
The link lists all the missions which you can click on to find out what you have to do.
To get the airship pass you either pay 500,000 to a NPC in Lower Jeuno in the building opposite the auction house or complete the Magicite mission 4 (which is listed in the above link) Unless you have help from high level players, don't expect to get to rank 5 before lvl 35 as a minimum as you have to fight 2 lvl 40+ NM's.
Before you ask the Kazham airpass (to the jungles for levelling lvl 25+) you have tog et keys from in all 3 nation areas, search wikipedia for the quest and you'll find out how.