Bards are the storytellers in game. Each bard will give you 3 different stories.
Quest will be enabled once you gain level and also finish the other stories.
Bard will ask you get certain items like (40 Immortal Hearts) once you got all of those it will ask you to kill 300 monsters (fix numbers) and then the quest is complete.
Stories are very important to finish since it will unlock Certain Recipe for crafting. These recipes may not be that important in the beginning but once you are done levelling and start building your items then that is where these recipes are used.
Check this new Article on why BARD QUEST IS IMPORTANT.