Quests

Click on a quest to get the orange icon on the right to light up. This means you've activated the quest and it will track you for the objective until you complete it.

Clicking on the orange icon will deactivate the quest. The quest will stop tracking you for the objective, but it will save its progress until you activate it again. No worries about dropping a quest, at least. This is important because you can only have 5 quests activated at any time.

When you complete the quest, the icon on the right will turn blue. Click on it to claim your rewards.

Note that expedition quests are evaluated upon successful completion of each expedition, so these quests must be active when your expeditions return. Crafting and repair quests are evaluated immediately, so there's no point in leaving these quests active when the timer is already counting down.

Daily quests are reset everyday at 05:00 JST (GMT +9). Weekly quests are reset every Monday at 05:00 JST. Both daily and weekly quests are deactivated when reset.

Bonus: HERE are the below Quests in Excel Format. This is so you can use it as a checklist and cross of the ones you've finished (using the Strikethough text option works for me). Made for myself and thought I'd share.

Double bonus: HERE is the above excel spreadsheet that has been updated to include the additional quests using an online format to use as a quick reference, also includes the expeditions chart.

Daily Quests
''All daily quests are to be completed within the day. Resets every 5AM JST.''

Weekly Quests
''All weekly quests are to be completed during the week. Resets every Monday 5AM JST.''

Expedition Quests (遠征)
For expedition quests, any quests should be active by the time your fleet comes back, or it won't be counted. It's not necessary for them to be active when you send fleets, but they must be when they come back.