Learn English – the action of riding a swing called

phrase-requestredundancy

Swinging is itself an action.

I believe I could say:

I swung on the swing.
or
I am swinging on the swing.
or
Do you want to swing on the swing?

Except all of those sound horrible to me.

We can also say:

Ride on the swing.

Is there a proper way of saying it?

Those may sound horrible, but they are nonetheless relatively common when talking about playground swings.

I went swinging on the swing yesterday.

Such redundancies are sometimes inevitiable when the same word functions as both a noun and verb. Here are some other examples that sound just fine:

I had too many drinks when we went out drinking last night.
I went fishing but caught no fish.

That said, these may be inevitable, but they aren't necessarily unavoidable. Such sentences can usually be rewritten if the redundancies bother you:

I had too many drinks last night.
I went fishing but caught nothing.

In the case of the swing:

I was playing on the swing yesterday.
I was using the swingset yesterday.

Sure, you could say:

I went swinging yesterday.

but in the case of swinging, you might want to clarify that (see Collins #3).