Mushroom are pretty easy. Plastic bag, a little water and heat or cold pack in the box(depending on where you're sending them). Next day shipping is best, but I've had shrooms survive after 3 or 4 days "lost" in the mail.
I've had a lot of success in the past with packing them in a cardboard box and some newspaper for a little insulation around the bags. For extreme temps, a styro cooler may be needed.
Maybe someone else can chime in with a better method?