There is more to the Bronx than just the Zoo and the "bad neighborhoods"
IMHO You get MUCH better Italian food in the Arthur Ave area (Belmont) than you ever would in Little Italy (Manhattan)
Then there is the home of the greatest Baseball team of all time.
The NY Botanical Gardens is awesome.
City Island is fun too (though it's a bit out of the way for a day trip)
Wave Hill is gorgeous.
There is more open parkland in the Bronx than any other borough. Drive through Pelham bay park and try to convince yourself you are in NYC.
I would recommend Coney Island. See if you can't catch a baseball game--The minor league teams will be cheaper: Staten Island Yankees or the Brooklyn Cyclones.
See if you can't arrange your schedule so you are in NYC for the 4th of July. Hit Coney Island for the Hot dog eating contest, then head to the west side of Manhattan for fireworks.
If in nyc in June see if you are around for June 23--catch the Mermaid Parade
For art, the major museums are nice enough, but the galleries and arts centers have more contemporary things. PS1 (now a part of MoMA) in Long Island city, Queens is a VERY short subway ride from manhattan plus it puts you closer to where the real art scene is in the city (it left Manhattan decades ago).