When the player first meets them, the Sprites speak in Pig-Latin. Giving them gifts will help them speak normally. The main Harvest Sprite is always found in the main cave, while the other two live in their little cave (the hole in the wall in the cave). If you go into their cove, you will notice another small hole which (apparently) connects to your barn (and explains how they are able to feed your animals during typhoons).
If you get a high enough affection with the sprites the two deeper in the cave will reveal their names: Arthur and Billy. The main sprite is Carlos. An interesting note is that Arthur and Billy both will appreciate Rare Metal and Moonlight Stone as gifts, while the Carlos doesn't care about those. Arthur is (apparently) a good cook, while Billy is not.