Being new can be frustrating at times.
To turn the layer's visibility on and off you need to set the Visible property on the Layer to false. To get the correct Layer object, you need to call GetLayer("Layer 1") on the LayerMgr. To get the LayerMgr object, you need to call GetLayerManager() on the ModelDoc object.
Here is some sample code.
Thanks Jesse, that's exactly what I needed!