How can I protect a burn from sun exposure?

I received a second degree burn on my hand and forearm two weeks ago. There was extensive blistering on my arm. I was treated at a burn clinic. The doctors mentioned that if the skin gets sunburned, it will be permanently scarred deep red. How much exposure to sun is safe. I have noticed darkening of the skin and wonder whether it is due to sun exposure? Am I risking permanent scarring by exposing it to the sun? I have been covering the area with a thin layer of moisturizer and a thin layer of gauze, but was told not to apply sunscreen yet.