When you have decided what mission you want to attempt simply tap the Mission icon on the Map where you will then be presented with a screen where you choose a Council Member and up to Survivors to make up your party. At this point to give yourself the best chance of being successful it is important that you make a note of the different levels and abilities of the Survivors and select the best party possible for the Mission.
Once you have put together your party press 'Send Raid' to deploy them. Once they reach their destination they will either succeed in the Mission and earn rewards or failed and sent back home to recover from the defeat. Even if you are unsuccessful you may still receive lesser rewards.
You can quickly travel between Missions which will fulfill daily quests or other assignments by tapping the 'binoculars' icon on the right side of the map.