I am writing this post mainly because I have received 20 emails asking this question in the past few days. This post will be a reminder to add this information to my Quick Reference Guide, but in the meantime, it can help players who have a problem with this.
First, you need to ask Gotz to build a Mushroom Shed for you. When the Mushroom Shed it built, you will have three options to grow mushrooms:
1. You can buy Shitake Seeds from Vesta (in the evenings when her storeroom is open for business);
2. You can gather Matsutake Mushrooms from the wilds in the Autumn season, then convert them into Seeds (Spores really) using the Seedmaker;
3. You can gather Toadstools from the wilds in the Autumn season, then convert them into seeds (spores) using the Seedmaker.
Once you have a bag of Mushroom Spores/Seeds of ANY variety, you must do the following:
1. Throw a piece of Wood Lumber onto one of the wooden pallets in the Mushroom Shed;
2. Throw the bag of Seeds/Spores onto the Lumber.
You will see tiny spores or seeds in the piece of wood if you aim your throw properly. You then need to water the spores daily for MONTHS in order to obtain a fully grown Mushroom. Each of the three varieties has a different growing time, but all three can be harvested at three different stages:
You need to ship one of each size and variety to complete your Shipping List. Moreover, if you look at the Item Guide in the Farm Menu, you will see that you can grow and ship 99 levels of each type and size of Mushroom.
To grow upgraded Mushrooms, you must throw more than one bag of seeds/spores of the same level onto the wood on the pallet in the Mushroom Shed. Five or six bags of seeds (spores) usually will guarantee that you will obtain the next level of crop in at least 8 squares when you plant 9 squares in the soil. In the case of Mushrooms, I never have failed to obtain the upgraded level even when I threw only 4 bags onto the wood.
BE WARNED: Upgraded Mushrooms can contribute to the dreaded 'freeze glitch' that can corrupt your data and destroy your game completely. I personally would advise players to grow only Level 1 Mushrooms of each size and variety. Doing so will complete your List.