In a bowl, combine the sugars and blend in the shortening. Add the egg and beat well. In another bowl, combine the flour, baking powder, baking soda, and salt. Slowly incorporate the dry mixture into the wet mixture. The dough will be on the dry or crumbly side.
Cut up the cherries into small pieces, but save one for garnish. Add the chopped cherries and pineapple and mix well. Line a cookie sheet with parchment paper, and drop 1 tablespoon of dough per cookie about 2 inches apart. Garnish each cookie with shredded coconut and a whole or half cherry on top.
Place in the oven and bake for approximately 15 minutes. Remove from the oven and serve.