Moma is worth a visit, but for a more lively Art experience ánd for free, one should visit Chelsea. A colony of 400 Art Galleries, showing the best in contemporary art. Running straight through is the high line, a 1 mile long linear park, build on a former railroad-spur, worth the walk.
Erie Basin is a really wonderful little shop full of old, creepy and beautiful things, such as Victorian mourning jewelry (some with hair in it), Freemason masks from the 1900's, tiny children's rings from the 1700's, and all kinds of other treasures. It is really as much a museum as it is a store, and all of the items have a haunting and unique aura.
City Island is a seaside town in the Bronx. Known for its fishing, seafood, and main street it is a self-contained working-class haven. The buildings range from late 1800's houses to 1960's storefronts, and you can smell the ocean everywhere on the island. Restaurants and local pubs dot City Island Ave, and even the few empty storefronts stay vibrant.