Diamonds are used to enter the Floating Isle Tower. It costs one diamond to enter the tower.
Three Diamond Shards can be combined to make a diamond at The Synthesizer. Diamond Shards can be either found in secret dungeons or bought from the secret shop in Freigold Town for 5000. To access the secret shop, keep swiping right.
Alternatively, diamond shards can be sold for 1500 and diamonds can be sold for 3000.
Diamond Shards can be found at the following locations. If you restart the game using the "Restart as Current" option, you can recollect the shards.