JNB's reference to the bicycle wheel is a good suggestion. Take a front wheel from a ten-speed and hold it by the axle. Have someone spin it, and then turn the whole assembly this way or that using the axle. You'll immediately feel the gyroscopic effects.

Some children's museums or science museums have hands-on displays of gyroscopic effects using a spinning weight that you can manipulate.

Pick up a children's toy gyroscope and play with it a little. Spinning, the gyroscope is rigid in space, which is the property we use in mechanical gyro instruments (attitude). When the gyro is displaced, its motion is predictable, and it's that motion that's used to establish rate rate of turn. Gyroscopic effects are applicable to any spinning object, the most obvious of which is a propeller.