Use MegaBus or BoltBus to NYC. FungWah has had an unfortunate number of accidents. It's about a 4 1/2 hour ride.
This is one Boston hostel, centrally located:
http://www.bostonhostel.org/
Things to Do:
Freedom trail (historic sights)
Duck boat ride (fun orientation to the city)
North End (Italian restaurants, bakeries, cafes)
Aquarium, Museum of Fine Arts, Gardner Museum (if that's your thing)
explore harbor islands by boat
whale watch
Faneuil Hall Marketplace (shops, food, bars, street artists)
Swan boat ride in Public Garden
South End: chic restaurants and performance space at Center for the Arts
Harvard Square, Cambridge (Red line T subway) Friday and Saturday nights esp. have street musicians, jugglers, etc.
clubs on Landsdown Street
clubs in alley off Boyston Street, near corner of Tremont
Middle East (club) in Central Square, Cambridge (Red line T subway)
Johnny D's (club) in Davis Square, Somerville (various live music) or Burren, Irish pub, Davis Square (Red line T subway)
Independent, Irish pub, Union Square, Somerville
day trips to Gloucester/Rockport; Newport, RI; Salem; Newburyport/Plum Island ; Lexington/Concord/Walden Pond; Cape Cod