You'll get more quests as you get higher in level. I believe it's 10 to activate the second part and 15 for the third part. Do some leves, do other peoples leves, kill some monsters, try out other jobs, craft, etc.
To use chocobos first you must reach level 20, then you have to run to Jeuno and start the quest. Trade the chocobo the correct greens every hour, for six hours, to get your license.