To start Ava's missions you'll have to go to Sanctuary III and head to Ava's room, which is located on the top floor at the bridge between the main area and the stern of the ship. On the way to the control area of the ship, you can turn right into Ava's room. There will also be a Yellow "!" leading into the room.
There are 4 missions to complete and each mission takes you to a brand new map. Each map is extremely linear, so there isn't much to explain. Each mission follows very similar story-beats:
-Meet with a character
-Have them explain the mystery
-Follow along with the linear map, taking out enemies
-Find clues
-Progress through the map, taking out enemies
-Conclusion
Once a mission is over, you will have to go all the way back to Sanctuary and back to Ava's room to accept the next mission. Each mission takes you to a different planet. You will need to have completed the main story in order to access the fourth and final mission.
The order of planets that you'll go to are as follows:
-Promethea (Mission 1)
-Eden-6 (Mission 2)
-Pandora (Mission 3)
-Nekrotafeyo (Mission 4)
NOTE: This can be buggy and you won't be able to accept the quest, as the prompt for Square to accept it won't appear. To fix this, either jump around or stand as far to the right as possible.