While in a park go to a coaster and select ride and you will ride it then press the x button and you will shout. This has not been tested on all coasters and it will not work if you have done this in a coaster from the coaster builder menu.
Press Square, Circle, Triangle, Square, Circle, Triangle, Circle. Then press Square at the 'Mission Selection' screen to complete the highlighted mission and unlock the next mission. Every time the Square is pressed on a highlighted mission the rank of the completion increases. The rank starts at Bronze, then Silver and finally Gold. If you want to reverse your progression on a mission you have to press Circle at the 'Mission Selection' screen and it will remove the highlighted mission medal.