Similar to Drink quests, Hot Spring quests are side quests available in the hot spring. They are unlocked as you obtain rank in either the guild or village so just do all the quests you can. Each one of them permanently increases the Health and Stamina bonus you get before quests by simply bathing in the hot spring. In Multiplier, the player with the highest hot spring values will have his hot spring water used.
Also like Drink quests, Hot Spring quests affect my recording directly so they will be posted as they are completed.