On a mushroom island, you should get before you go:

-A bowl. The only food avaliable is mushroom soup, so bring a bowl for food.

-Saplings,bonemeal and dirt: Break a bit of mycelium, replace it with dirt, and put a sapling and grow it with bone meal. If you leave a sapling without bonemeal, the mycelium will take over the dirt and un-plant the sapling.

You do NOT need (These are a waste of inventory)

-A sword. No hostile mobs spawn on a mushroom island,even underground, so you don't need worry about creepers. But if the island is attatched to another biome, monsters spawn there.

Hope you have a nice time with the mushrooms!