On the main Missions screen (after selecting "Missions" on the main menu), you will be presented with 4 different types of missions for each Zone. Taking the first Zone as an example, these are Campaign, Manhunt, Contract, and Boss missions.
You may select the mission type that you would like to play here. Once a mission type is selected, you will be shown the description and reward of the mission on the right of the screen. Tapping the “Play” button in the bottom-right corner will take you to the Weapon Screen where you can choose the weapon you want to use for the mission. When finished, as long as you have the indicated amount of Energy to proceed (shown on the "Play" button), tapping the "Play" button will begin the mission and consume the mentioned amount of energy. If you do not have enough Energy, you will be taken to the Get Energy screen to get more.
At the top-right corner of the Missions screen, you will see the zone name along with arrow buttons on either side to allow you to change the zone.