In my opinion:
NYC is like Boston, San Fran, London, Paris and many others in that it's suited to walking.
Get your hands on a Manhattan bus map (free at tourist centers) and Subway map. The MetroCard is good on both and there are free transfers between lines as in many cities.
You want to fill your MetroCard in $10 increments. You get additional credit for every $10. NOTE: The MetroCard is time stamped and up to 4 people can swipe the same card in the Subway or on a bus. After the 4th person the card won't work even if it has $1,000,000 on it.
DO NOT go for ANY "Hop ON-Hop OFF" bus. That's not a tour. It's a routed bus and you'll spend your time waiting for the huge, lumbering bus to come.
A relaxing tour option from your link is either the Brooklyn or Bronx tour. With those you can sit and actually see things not often included on TV etc.
What you do is pick an area or neighborhood. Go there and walk about locally. Also, decide what you want to do ahead of time. You see 4, 5 and more people standing on the sidewalk unable to decide what to do next. If you're in a party then show a little maturity and independence and be willing to break up and meet up X hours later.
Remember, many will be embarrassed to say they want to go shopping (a HUGE tourist thing to do everywhere in the world) while others might be reluctant to select a small museum (or big one) for fear of boring the others.
If I get any response from the OP- or anyone else - I'll go into detail. PM me WhyMe and I'll send you a Bus map.
The Staten Island Ferry is free. Take it out to S.I. and back. You get close to the Statue of Liberty. Going out to the Island involves a ticket and separate boat ride which is OK but the line at Castle William (where the tickets to both St. of Liberty and Ellis Island are sold) can be long. The Ferry, Wall Street, The World Trade Center Site and Memorial, Fraunce's Tavern (where George Washington threw a Good-Bye party for his Generals after the Revolution) and other stuff are all in that one neighborhood. You can walk from one to the other.
More upon request.
Personal note: Friends from England came over with 2 years notice. I told them repeatedly to bring BROKEN IN walking shoes. They didn't AND they complained the whole time. What a bummer it was to be with them. IF YOU DON'T WALK be real specific with what you want to see and go "item-by-item".
Are you interested in...
Museums: (large and small) The Metropolitan Museum of Art is big. The Museum of Holography is small. The Brooklyn Museum has the largest collection of Egyptian art in the world (maybe second to the Egyptian museum in Germany). The Museum ofPhotography is good for changing its exhibits. You can also use public transportation (TNJ bus) and go to Thomas Edison's laboratory in Edison, NJ. That place is cool.
Shopping: Bloomingdale's is a Mecca to many. Lexington Ave. from 57th to 96th St. makes a fine "shopping walk". People love SoHo (South of Houston St.) for pubs and shopping.
Music: Lincoln Center's "high end" stuff or the bars in Greenwich Village.
Theater: Go to the TKTS booth at Duffy Sq. (the top of Times Sq. at 47th St.) as mentioned above.
TV: Go to Rockefeller Center (NBC) during the morning shows and stand outside the window and look like an imbecile waving at the camera.
WALK: Definitely walk the span of the Brooklyn Bridge
Take the 2,3 subway (red lines on the map) to Clark St. in Brooklyn to see the "old Brooklyn" of Arsenic and Old Lace. The view is great.
