General Quests are given by any other NPC in Madrigal and involve sub-stories where objectives are completed for rewards. General Quests come in two forms, Single and Chain.
A Single as the word says has only one part.
A Chain has many parts, all must be done in order. Often items from a previous stage will be required to continue quest. So keep in mind that you might need the item you received from a quest stage. Don't throw it away if you aren't sure if you can or cannot use it later on. A quest has a giver NPC and a receiver NPC. In general, the receiver will be the giver of the next stage. Thus having a chain effect.
When an NPC has a yellow exclamation point on his head. He has a quest for you
When an NPC has a gray question mark on his head. He is waiting for you to finish the quest.
When an NPC has a yellow question mark on his head. You are ready to hand in quest to him.