Tarcore is a material used to level up tarstones from level 2 to level 3. This material can be obtained by leveling up other Tarstones.
Try using and leveling up different tarstones to get Tarcores. You can then use those Tarcores to level up more Tarstones.
After leveling up tarstones, tarcores will be dropped at its mouth. Interact with the front portion of the Tarforge to claim acquired Tarcores.