2 questions. 1: what are all the kids' quests? 2: how do they find them out?

John’s was to free the fireflies, which he did by removing his planet of oil

Rose’s is to play the rain and bring life back to her planet’s ocean

Jade’s was to stroke the forge and breed the frogs

Dave’s hasn’t been stated directly in canon, it could relate to the Forge or the legendary blade Caledfwlch.

The Alpha kids’ has not been stated.

They learn their quests by learning their planet’s lore and by speaking with their denizen.