If you would like the soup smoother, feel free to hit the finished soup with an immersion blender.
I highly recommend using half and half if possible, otherwise, you can substitute it with whole milk. Note that if you do substitute with whole milk it may be a little thinner.
Shred the cheese fresh from a block yourself instead of using prepacked shredded cheese. Prepackaged shredded cheese contains preservatives that prevent it from not melting as well and can also make the soup grainy.