This is very simple to unlock.
Once you start the game and enter level 1, collect one crystal (which looks like an emerald) and the achievement will unlock.
Complete the level to also earn:
As reported in the Xbox Achievements forums (and confirmed by Shinnizle), you can play level 1 three different times to get the other achievements (Credit goes to bLaKgRaVy for this.)
https://www.xboxachievements.com/forum/topic/484388-mushroom...