This is a list of some of the missions you can find in game, in the approximate order that they appear. The different colors of the mission icons represents the overall difficulty of the mission in relation to your current power level.
Play with others
Selects a random match for you to join, and will auto-fill with other players.
Fight the Storm
Find the ATLAS, and defend it until the timer runs out
Ride the Lightning
Find Lars, and defend his Van through waves of enemies
Retrieve the Data
Find the landing zone for the data balloon, and defend while the data is downloaded
Rescue the Survivors
Search the map to locate Survivors and save them from various hazards
Destroy the Encampments
Find the active Husk encampments around the map, and kill the enemies to destroy them
Build Radar Grid
Find the Radar towers around the map, and build them per the outlined blueprints
Fight Category 2 Storm
Fight the Storm, but with more ATLAS(es) to defend
Repair the Shelter
Find the Shelter location on the map, locate the parts, and defend against waves of enemies (highly recommend bringing a friend)
Mutant Storms: While Mutant Storms are not a mission on their own, they are a modifier that can be found on ANY mission type.
Offer modifiers for enemies and heroes, to make the mission a bit more difficult (eg. heroes share damage, husks explode on death, etc) which can be seen in the description of the mission on the left side of the screen: