At level 22 you can sign up for a grand company, the storyline involving what is happening with the Meteor and the Garlean conflict can be started then. You can enter a grand company via the second option on the entrance to the market wards. You'll need to sign up for all three and hear what each company has to say as a Recruit before picking one to advance your company career. Be warned that this storyline can only be done until 2.0. After then, they plan on introducing a whole new replacement storyline since the majority of the cut-scenes here are warning about the whole upcoming cataclysm-end-of-the-world-meteor coming in a few patches.
At level 20 you can start undertaking class quests. This is done at every guild corresponding to the class after level 20. There are three of these. One when your class gets level 20, one when class gets level 30 and the last current one when it hits 36.
At level 30 you can start undertaking job quests. You get the job and each ability for each quest you complete in the line. The job quests are done at level 30, 35, 40, 45, and the last one is done at 50.
The primals (Ifrit and Moogle) are able to be done after you complete the level 25 "Easy" version of Ifrit in the company quest "It bleeds with Fire".
The main scenario quests are done historically at levels 10, 14, 20, 26, 30, 34, 38, 42, and 46. I know they lowered the levels of some of these, so this particular information may not be correct.
Edited, Mar 29th 2012 3:04pm by UltKnightGrover