Here is a list of all the quests that are called Main Scenario Quest. It's a long linear quest line that tells you the story of the game and the quests are marked with a flaming quest icon.
http://ffxiv.gamerescape.com/wiki/Ca...Scenario_Quest

If it's your first time leveling, it's highly recommended that you do those quests, because they introduce you to the game and unlock things. If you don't reach a required level for the next step in the line just by doing the scenario, you can do side quests (regular quest icon) and fates in the area. Other good ways of getting exp are guildhests, dungeons, hunting log (enemies with a yellow icon on their name) and levequests (repeatable, although best when saved for crafting). The challenge log gives you free exp if you complete an objective, for example do 20 fates, and you get a notification if you are close to finishing one.

These other things are usually only required if you are leveling a second time with a different class, and on your first time leveling it's best to follow the main scenario. If you dislike your class choice, you can pick another and catch up to the main scenario's level with the side objectives I mentioned. Don't forget to do the class quests every 5 levels, found in the guild for each class. At 30 you can turn your class into a job, making it more effective.