Arthur Avenue is a mainstay of the Bronx. Since the early 1900's til today, it's been the hub of the Italian-American community that calls the Bronx home. Many family-owned, traditional businesses dot the main avenue, and the food culture is rich and boisterous.
Coney Island is the best for everything - the beach, the rides, the beer - but mostly for the Skee Ball. It's incredibly addicting, affordable and satisfying (depending on your aim).