When "swing" is a noun you would normally say PLAY. Let's go to the park and play on the swing. When it is a verb, you could say: Look! This child is swinging so high! Let's go for a swing in the park. You would use "ride" for a horse or a car. Can you ride the horse? Do you want to go for a ride in the car? :-)

You could say 'play', but 'go on' is also commonly heard. "I'm going on the swing in the park"