To upgrade your gear in Lost Ark you need to obtain the materials needed first. For tier one gear you need to get Harmony Leapstones and Harmony shards. Then you also need Guardian stone fragment armor items and Destruction stones fragment for weapon upgrades.
The process is pretty simple you need to go to the gear honing NPC, there is one located here in Vern castle.
After this simply talk to the gear upgrade NPC select gear honing and the item you wish to upgrade and click upgrade.
After this you click on gear honing, then you can see what percentage you have of succeeding in the upgrade. Its also possible to add items to get a higher success chance, like the item Star’s Breath.