From the pictures, it really looks as if the redness around his gills is part of his natural coloration. It's really common for fish that are under stress (i.e. when they're moved, as in your case) to wash out. As they get more comfortable in their new surroundings, they color back up.
However, just to be on the safe side, could you post more info about your setup? How big is his tank? Is it heated and filtered? Can you post your water parameters (temperature, pH, ammonia, nitrite, nitrate)?
A betta thrives at tempertures between 76 and 82 degrees, so ideally you should have a heater for the tank. Low-flow filtration would also help keep the tank clean and would provide a place for beneficial bacteria to grow. Your water parameters are all very important for the health of your fish, so having a good liquid test kit like the API Freshwater Master Test Kit will let you monitor these levels and keep your fish healthy. You can also take a sample of your water to a fish store for them to test, but make sure they give you exact numbers for pH, ammonia, nitrite and nitrate. In the long run, though, having your own test kit will save you a lot of trips to the store (as well as save you a lot of time, money, frustration plus the lives of a lot of fish, to be sure).