You’ll receive notice of this quest when you first enter the Stardew Valley Museum & Library, located in the southeast corner of Pelican Town, on the other side of the small river that meanders through town. The quest doesn’t actually trigger, however, until you enter the building with an Artifact or Mineral in your inventory.

Artifacts and Minerals are listed under the Collections tab of the menu. Both are collectables that you’ll find by going on scavenger hunts throughout Stardew Valley. How you go about this depends on whether you're targeting an Artifact or a Mineral:

  • Artifacts: Wander around on the dirt paths between occupied areas (the road between your farm and Pelican Town is a good place, as is the beach south of Pelican Town) and watch for small, wriggling things that look a bit like worms sticking out of the ground. Use your Hoe to dig the spot up and you’ll find an item. If you’re lucky, you’ll receive an Artifact. If not, keep looking. You’ll find one eventually.
  • Minerals: Pull out your Pickaxe and smash rocks until you find a few Geodes. Take the Geodes to Clint, the blacksmith. He can bash them open for you, which may net you a Mineral. (You can also enter the Mines to find Minerals, but Geodes are a quicker, safer way of getting the job done, and you can find them from the first day onward.)
Once you’ve located an Artifact / Mineral, head to the Museum. Upon entering you’ll hit a cut scene with Gunther, the proprietor. Once the scene is done, speak to him at the front desk to hand over your Artifact / Mineral. You won’t get anything for the donation, but you will receive 250 gold for completing the quest. Check the Archaeology Journal entry to claim your prize.