Like so many other "Games" games, European Games is primarily a joystick-waggler, with some precision elements thrown in. Taking place over five or six disciplines, depending on platform, you generally go left to right (or hit the right keys) with your joystick on those judged on speed, while pressing the fire button once a metre reaches its top (as in golf games) is the general method for disciplines.
Throwing is handled by releasing your hammer when the power bar is at the top, with additional rotation control on the Commodore.
Swimming involves more joystick action, but you also need to keep an eye on your air supply. Going up for air slows you down, though.
The long jump involves yet more joystick action, until you reach the touchboard. Then you need to press the jump button at the right time, optionally choosing jump angle.
Rowing is like swimming, but without air.
Weight lifting involves more power bar pushing, keeping the bar high constantly and lifting the weight before the time runs out.