You get faction assignments from the two leaders in the Helicarrier and Ant Hill outposts. You can find them by pressing up on the D-pad.
You can pick up 8 per leader, per day, for a total of 16 per day. These quests have a little bit of everything in them, kinda like Destiny's daily bounties. They range from simply completing missions, to killing certain enemies, to opening chests.
In the Helicarrier, the Shield faction leader, Alisande Morales is located in the top of room with the war table.
In the Ant Hill, the Resistance faction leader, Sarah Garza is located in the big room at the wall directly behind the hero changer when you walk in.