To clarify, there are 3 types of quest icons. Theres the oval, the oval plus, and the comet. Oval quests are optional - youll get some exp, maybe an item or two, and some lore but thats about it. Oval plus quests are optional as well, but they unlock more of the game for you. New dungeons, new commands, new anything. This quest unlocks the Gold Saucer for instance. Follow the link to see the Oval plus icon. The comet icon quests are the MSQ quests that people mention. This one here is the first one for a gladiator(and again, the link shows the icon in question). The MSQ(main scenario quest) line of quests tells the story of the game, and unlocks a LOT of things you need/want to be able to do. Hopefully that helps a little.

Bonus tip! You need to complete the level 10 class quest from the class you started as to unlock the armory system. This is the ability you have to switch classes and become something new(or just to become everything). So make sure you do that before making the trip to Limsa so you dont have to go back.