A youngster can gather vines and some other things. You can start there. Then just try to gather everything you can, as aoh said. You'll need those things for later puzzles and trades and crafting anyway, so build up inventory. During a recent holiday event (which is now over), a player needed something like 50 or 80 units of sand to fully participate in the event. Some resources can't be gathered until you complete some of the puzzles, and a few are gathered as part of farming. All but one of the items on the first three pages of the crafting hut can be gathered. The exception is something found in the earthquake cracks.

I don't believe a villager gets gathering experience from gathering mushrooms or fireflies.