Knowing that there are problems with this quest if one should inadvertently pick up the items before getting the quest, in my brand new game, do you think it's a good idea to go get the quest right off before I do much dungeon delving and accidentally run into the quest items?

If you have all the official patches, you shouldn't have problems with instruments being stuck in your inventory after giving them to the bards, even if you pick them up before you get the quests. Confirmed at least with Finn's Lute for me.