32 c means your tires are 32 mm wide. You shouldn't need new wheels just new tires, but depending on your rim its not clear how narrow you can go. The best is to bring your bike to a LBS and ask. 32 mm is 1 1/4", 28mm is 1 1/8", 25 mm is 1", 23 mm is ~3/4" for comparison sake. You also want to get a tire that is smooth, light and goes to high pressure. Tires can cost anywhere from $15 to $60 each depending on how fancy they are.