As a lancer, you want Lancer to level 30 and Marauder to level 15 to unlock the Dragoon Job; though leveling pugilist will also give you usable abilities on Dragoon. As Archer, you need level 30 Archer and level 15 Pugilist to unlock Bard, though a few abilities come from Lancer.

To questing, Follow the Main Storyline Quests (the ones with the Fiery Quest Icon) and do all the quests that are around it as you go through. Some of the MQ will be instanced battles, some simple kill/gather quests, some are just go talk/listen to someone, others require you to go through a dungeon. The dungeon ones require a 4-man party 1 tank, 1 healer, 2 dd/dps: queue times can be pretty intense so see if anyone needs or is willing to run with you if you can.