FFXIV quests come in 3 main categories;
Main Scenario - which covers the quests you start with when entering a town that follows through the storyline. When you reach the rank for the next quest you will get a notification via linkpearl.
Class Quests - which covers the 3 (at this time) quests for each class at rank 20, 30 and 36. Each quest builds on the lore of that class and allows you access to purchase higher rank skills with guildmarks. After joining the guild and getting the linkpearl when each quest is ready you will get notification.
Side Quests - which covers the sundry quests started at various NPCs throughout the 4 main regions which involve talking to people, killing mobs or collecting items for a reward of gil or items. While not directly linked to the main scenario quests, some offer insights to possible future events like Grand Companies while others flesh out the lifestyles of the people of Eorzea. The rank requirements for sidequests covers from r5 to r45. Starter NPCs and any other NPCs involved will have a question mark bubble over their heads. Some quests do have pre-requisites for other quests in the same or different area that have to be completed first