The Destruction levels require you to hit all destructible objects found on the track. Drive directly into them to destroy the item. Make sure your progress bar is at 100% before reaching the end of the track to make sure you have hit everything.
Precision levels
The Precision levels require you to pass a set number of checkpoints and park at a designated location. You will need to be patient and use slow speeds in order to navigate the obstacles. Touching any obstacle will result in failure. However, you cannot take too long to complete the level in order to beat the times required to earn stars.
Stig levels
The Stig levels involve you racing against Stig. They have similar layouts to the Precision tracks. Avoid hitting the cones which will cause you to lose five seconds from your current time. Reaching checkpoints fast enough will reward you with time boosts. Crashing into an object will end your attempt.