To progress through the game's story, the main thing is to complete all the quests from the list on this screen. You don't have to worry about the indicators next to the character names.
When all the quests are completed and you have free time. you can try to complete recurring character quests (invite them to fight monsters, talk to them, or just fight with them). This will improve some of your character interaction skills.
It's a pity, but there are no such quests for Edgar yet. You can't even treat him to beer. This is the main drawback of the game right now.