In Lost Ark, the Harmony Leapstone is required to progress the main story from Shushire. You can use them to upgrade all gear, including weapons. They are normally awarded for completing Chaos Dungeons, but you will regularly see them as log-in bonuses and event rewards.
How to farm Harmony Leapstones
The only way to farm Harmony Leapstones is by clearing Chaos Dungeons or Guardian Raids. You're sure to earn some every time you repeat raids and dungeons, but they're not very plentiful. Still, Chaos Dungeons and Guardian Raids are the only way to farm Harmony Leapstones with confidence.
You will only receive many Harmony Leapstones the first time you clear a dungeon. So continue to level up your gear to get a bonus in Harmony with the higher level dungeons you clear out.
You can find the icons for Guardian Raids and Chaos Dungeons above.
As a reward for logging in every day, you will receive many Harmony Leapstones, but it depends on the day. Crystals can also be obtained as a daily bonus, and you can use them to purchase more Harmony Leapstones. It's best to keep grinding for these gear-honing items, as there's no set way to get a lot at once without paying for it.
For more information on Lost Ark, check out Can you make a Guardian Stone Fragment from Guardian Stone Crystals in Lost Ark? or What is Gear Transfer in Lost Ark? Right here on Pro Game Guides.