Lower East Side -- Russ and Daughters (foodie shop, lots of smoked fish)
The Pickle Guy, the Tenament Museum.
IMO the lower east side is fascinating, but may not be th ebest destination spot for a first trip to NYC. IMO better to include it on a walking tour of lower Manahattan (along with the East Village, the Village, SoHo, perhaps ).
South Street Seaport is a mall-kinda thing like Faneuil Hall in Boston and the Inner Harbor in Baltimore. Also more of something you might want to include as part of a walking tour but maybe not as a destination spot.
It's a pretty easy walk from Little Italy/Chinatown, through TriBeCa to WTC, past City Hall and the courthouses where they film Law and Order. Lots of very cool architecture. After you have walked around the WTC site, you can tour the World Financial center and the Esplanade and/or go shopping at Century 21.