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.
This is a pro tip, but also a summer tip, take the ferry from Lower Manhattan to Rockaway and spent the day. The beach is great and the area is impossible to gasp that it's 30 min from Manhattan (without traffic) - Eat as many tacos you can, cause it's the best in the state of New York!
When I first visited NY in 2010 the New Museum left a permanent impression. I love the architecture and the concept of the piled-white-shoe-box look-a-like building at the edge of the Lower East Side neighborhood. The exhibition program is appealingly different, and the bookstore in the foyer is not to be missed!