The pickaxe is an essential tool for playing My Time At Portia, and you will get a quest to create one near the start of your adventure.
To create a pickaxe, you will need 13 x Wood, and 13 x Stone. The problem is that you won't be able to mine stone since you don't have a pickaxe yet. However, both of the crafting materials can be found to the left of the entrance to Peach Plaza. So look there and loot the crafting materials, and you should be able to craft the Piackaxe.
When you have the materials, return to your workshop and interact with the worktable outside of the building. You can interact with the key "E" on PC, X on PS4, and button A on Xbox One. Now find the pickaxe and confirm that you want to make it.
And that's it! You should have your pickaxe by now, and you can now go ahead and mine as you please.