In order to trigger this quest you need to first gain access to the Mines. They’re located to the northeast of Pelican Town, a relatively short trip east of the Carpenter’s House. They'll unlock during your first week in Stardew Valley - just wait for a letter from JojaMart saying the way is clear.
When you enter the Mines and descend at least one level you’ll find plenty of stones to bust up. Crack them open and you’ll start finding all manner of Minerals, among other resources. Collect a few, go as deep as you want (you'll find ladders beneath rocks - you can even fight Slimes with the Rusty Sword you gained upon entrance), then head back up. Targeting the coppery-looking rocks displayed below will give you the best chances of finding lots of Copper Ore in a short period of time.
Go home and sleep. After napping for the night you’ll leave your house and find Clint, the blacksmith, waiting out front. He’ll give you plans for Crafting a Furnace. Furnaces allow you to force new items, among other things, and you’ll want one to improve your farm in general. You’ll need to following to building a Furnace:
- 20 Copper Ore - Found in abundance in the Mines
- 25 Stone - Found by busting up rocks; you’ll find plenty in the Mines as well
Create the Furnace once you have all the parts you need - but make sure you collect an extra five Copper, along with a piece of Coal. The final step in the quest is to create a Copper Bar with these resources. Interact the Copper Ore with the Furnace to trigger the smelting process, wait a few moments for it to finish, and grab the Bar out of the Furnace. This completes the quest.