Pokémon Legends: Arceus has two different types of quests — missions and requests. The difference between a mission and a request is their importance. Missions need to be completed in order for the main story to advance whereas requests are purely optional side content.
Aside from giving you new gameplay objectives, requests can also net you some nice rewards, which are especially great in the early game when you’ll need to stock up on extra resources for crafting. Missions will be given to your automatically as you progress, but requests are given to you in two different ways.
The first way to get a request is by noticing icons in the overworld. If an NPC has a request to ask of you, a diamond-shaped marker will appear over their head. Follow the marker and interact with them to hear out their request. Just speak to them again when you’ve completed whatever objective they’ve tasked you with in order to turn it in and get your rewards.
The other method of getting requests is by visiting Professor Laventon’s office in the Galaxy Hall at Jubilife Village. To the right of the office entrance is a chalkboard. Interact with it to see a list of additional requests for you to add to your quest list. You can check all of your missions and requests by pulling out your Arc Phone with the minus button.