Quick answer
To temper Tarstones normally, restore the Temper Tarstones function at the Tarforge by bringing Etching Needles back to Franz in Marrow Keep. Keep a Tarstone slotted on equipment you are actually using, kill enemies to fill its XP bar, then return to the Tarforge and temper it when eligible.
Unlock the Tarforge function
Current launch routes place the Etching Needles in the Glutted Mire / Magdalena-side progression. Bring them to Franz at the Marrow Keep Tarforge and the Temper Tarstones menu option becomes available.
The Tarforge has other upgrade parts for melee weapons, sidearms, smelting and the late upgrade cap, but you do not need all of them just to understand Tarstone leveling.
How Tarstone XP actually works
A stone needs to be equipped/slotted on equipment you are carrying for its XP to grow from kills. You do not need to trigger the active effect every time.
This creates a useful leveling trick: keep a stone equipped while doing normal exploration even if it is not the centerpiece of the current encounter.
Temper only after the bar is ready
When the Tarstone’s XP requirement is filled, return to the Tarforge. The temper menu will show eligible stones and the next material/coin cost.
Current launch guides publish exact XP/Tarcore thresholds, but those are economy values that can be patched. This guide intentionally tells you where the requirement is displayed in-game rather than pretending an August 20 number is permanent.
Do not max every stone you find
Tarstone resources are part of the same build-economy problem as weapon upgrades. Level the stones that create your actual combat loop, then use spare slots to train future options while you explore.