Below, I have listed some examples of different kinds of exercises that can be safely performed by children. Children should always exercise with adult supervision.

Stretching

Stretches should be performed for both upper and lower body after a brief warm-up. The child should hold the position for a few seconds. They should never stretch in a bouncing motion. Bouncing is not safe or effective. Some examples of stretches that should be performed before exercising are:
Hamstring stretch
Hurdler’s stretch
Quadriceps Stretch
Calf Stretch
Triceps stretch
Shoulder and chest stretch
Rotator cuff stretch

Calisthenics

Calisthenic exercises use the weight of the body for resistance. Resistance can be increased or decreased by changing body position or by doing more repetitions. Some examples of calisthenic exercises are:
Push-ups
Pull-ups
Bench dips
Crunches
Reverse crunches
Back extensions
Squats
Calf raises
Lunges

Tubing and Exercise Bands

Exercise bands and tubing are a safe and effective way to develop strength. Tubing comes in different lengths and tensions. When finding the correct tension you should be able to do at least 10 repetitions. Before increasing the tension you should be able to do 20 repetitions. Some examples of exercises that can be done with tubing and resistance bands are:

Seated shoulder press
Standing chest press
Side Bends
Lat pull downs
Triceps extensions
Bicep curls
Wrist curls
Squats
Leg curls
Calf raises