To Start will tell you who to talk to or what to do to start this quest.
The second section is what the NPC says when you talk to them to start this quest.
The third section is the text displayed in the More Info expanded box. It is usually a slight rewording of what is said in the previous section, and some editors skip this section entirely as redundant.
The last section is for any notes about this quest and includes whether or not the quest is Members Only, Repeatable, Job-specific and it's type, if any.
This will be ALL of the goals of the quest, as if they were all revealed to you at once. Most quests only reveal one goal at a time and only show the next when you complete the first.
This will include items, if any, and coin, stars and Treasure Tickets.