Quick answer

For an ordinary weapon, collect Smithing Stone [1] in Limgrave Tunnels and use the smithing table at the Church of Elleh. The table handles early upgrades up to +3 for standard weapons; you also need runes. Check the weapon's upgrade screen for the exact materials and cost.

For an ordinary weapon, collect Smithing Stone [1] in Limgrave Tunnels and use the smithing table at the Church of Elleh. The table handles early upgrades up to +3 for standard weapons; you also need runes. Check the weapon's upgrade screen for the exact materials and cost.

One early collection route

  1. Find the Church of Elleh in western Limgrave and activate its Site of Grace. Pick up the Smithing Stone [1] on the anvil beside the merchant.
  2. From the northern edge of Agheel Lake, locate the entrance to Limgrave Tunnels and take the lift down. Rest at the grace inside before proceeding.
  3. Search the tunnel walls and side passages for glowing mineral deposits and collect them. Miners can be dangerous in groups; pull them apart and retreat to the lift or grace if needed. You do not have to defeat the tunnel boss to keep stones you already collected.
  4. Return to the Church of Elleh and interact with the smithing table. Select your weapon, read the required stone and rune cost, and reinforce if you have enough. Later upgrades can be handled by Smithing Master Hewg at Roundtable Hold.

If tunnel miners resist your slashing weapon, try a weapon that deals Strike damage and fight them one at a time; the community enemy reference reports resistance to slash and weakness to Strike. Check the weapon’s damage type first. (source)

Standard or somber?

Weapon upgrade screen asks for Use
Smithing Stone Ordinary stone of the displayed tier, commonly used for standard weapons.
Somber Smithing Stone Somber stone of the displayed tier, used by certain special weapons.

The two materials are not interchangeable. Limgrave Tunnels is an early source of ordinary stones, not a complete list of every location. Ore pickups do not simply respawn when resting; don't assume you can collect the same deposit repeatedly. If a weapon requires a different stone, inspect its upgrade screen before travelling.

Sources: Eldenpedia Limgrave walkthrough · GameSpot smithing stone guide · Limgrave Tunnels overview.

Sources