Each island has a set of main quests for each of the three divisions. Doing quests for one division will lower the reputation of the other two. There are also smaller tasks you can request in the management screen at any time. (There is about a 1 minute cooldown if you cancel a task). The main missions have a border around the faction's logo and the tasks do not have a border.
Doing missions and tasks for a faction will lower the reputation for the other two factions, so I found it easier to work on one faction at a time. It may be hard to ignore the other faction quests early in the game because income will be slow moving, but do what you can to max out one before moving to the other to avoid wasting a lot of time. When you run out of main missions, keep doing tasks until the reputation is maxed out.
The achievement unlocks when you reach the maximum reputation for Science.