The Smithsonian is in Washington DC, not NYC.
Most of the above suggestions are rather
far-flung and time-consuming, especially
if you're only here a few days. Some
other possibilities:
Rather than head out to New Jersey for
deli food, take the #1 subway train to
79th Street, walk to 80th Street to
Zabar's, assemble the makings of lunch
from their many offerings, then walk
west to Riverside Park to eat it.
Once finished, get on the #5 downtown
bus on Riverside Drive and let it
drive you down Broadway and 5th Avenue
past many of NYC's sites. It used to
go all the way down to Battery Park
but now it ends at 31st Street and the
rest of the trip is covered by the #55
to which you can transfer at 42nd Street.
If the idea is to get a sense of New
York rather than just tourist attractions
you can just wander the streets of
various neighbourhoods, where often
there are interesting features which
don't make the guidebooks. Some areas
are better than others for this, of
course, such as the West Village (west
of 7th Avenue, between Houston and
14th Streets).