First, you shouldn't really be touching your betta. I know some people like to do it, but it really isn't good for them. There is no telling what is on your hands, and you can rub off their protective slime coat.
Also, what is the texture of the spots? Are they fuzzy?
Please fill this out too so we can get an idea of his situation: (I know you just got him, so fill out whatever applies)
What size is your tank?
What temperature is your tank?
Does your tank have a filter?
Does your tank have an air stone or other type of aeration?
Is your tank heated?
What tank mates does your betta fish live with?
What type of food do you feed your betta fish?
How often do you feed your betta fish?
How often do you perform a water change?
What percentage of the water do you change when you perform a water change?
What type of additives do you add to the water when you perform a water change?
Have you tested your water? If so, what are the following parameters?
Symptoms and Treatment
How has your betta fish's appearance changed?
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)?