that is some deep marring on the paint. You should be able to polish out most of it but your going to have to take it to a professional who knows what he's doing.
Keep the sponge in the kitchen and use a sheepskin mitt to clean the car. If you have stuck on bugs, take a fleece blanket that you don't mind messing up, soak it in a solution of carwash soap (the stuff you normally use) and drape it over the front 1 hr before you start washing. The bugs will rehydrate and be easier to remove.