Seth Kugel at the New York Times has pulled together a handy tour that covers Manhattan from top (the Cloisters) to bottom (Bowling Green). The trip, which takes 3-1/2 hours, relies mainly on using the city's public buses, with a bit of subway and some walking added to the mix.
If the idea works for you, check out our similar approach in A DIY Tour of Berlin. Along with a slide show, there's a clickable map showing vids of places you can reach on the M29's bus route.
And if it's Paris you're headed for, check out Gridskipper's rundown of the best sightseeing routes there.