Being a native New Yorker I second the Staten Island Ferry. Before you do anything, go to a Metro Pass vending machine in Grand Central Station and buy a two day unlimited trip subway/bus pass. Then walk up to Fifth Ave and take the number 1 bus to the end of the line. Hop on the Ferry and take all the pictures you want for free. The number 1 bus should be on everyone's to do list. It runs past many of the city's most interesting sights and it is really a pleasant ride.
After you get off the Ferry walk over to Chinatown and have lunch. Lunch for two, with really terrific food, should cost you less than $10. While you are there tour that area. Walk up a little bit to Little Italy and the rest of Chinatown, it has spread everywhere. Buy a wonderful Italian Pastry at Ferraro's. Don't worry about the line it will move quickly. If you want to walk up to the village or to South Street both are doable but better yet take advantage of your bus pass to take buses between venues. It will save your feet even if you are only riding a few stops.
After you have finished Downtown, take the subway or the bus uptown. If you take the First Avenue bus you will go through NYC's lower east side, midtown and the UN. Get off the bus at 34th Street and take the number 34 bus across town to the Empire State Building. Buy your tickets online before you leave your house. Do not go on that stupid ride.
Take the Madison Avenue Bus uptown to 50th Street. Walk one block west to Fifth Avenue. Take in Rockefeller Center, St Patrick's Cathedral, Sacks Fifth Avenue, Radio City, the Museum of Modern Art, the Museum of Television and Radio or any combination you can fit in.
Now get on the number 42 bus. It goes down Fifth and turns on 42nd Street. Get off on 42nd and 7th Avenue and look around Broadway. Stand on line at the new Tickets office and get half priced tickets for a Saturday night's show. Before you go into the show head down 42nd Street toward 8th Avenue. On the southwest corner, almost on the corner, will be the most outrageous Mc Donald's you have ever seen. It is in a converted theater and you sit in the orchestra, the loge or the balcony to eat your meal. The meals are almost the same price as any other Mc D's and the food is the same. While in there sit on the top floor and watch 42nd Street and Broadway go by.
After the theater go get a snack at Lindy's. Make sure you get the cheesecake. Lindy's is a Brooklyn institution but has a satellite restaurant right on 45th and Schubert Alley, in the heart of the shows.
The next day finish midtown and head uptown on the bus or subway to the museums. Of course you could shop instead, but I would rather do the museums myself.
Take care. I hope you have a great time.
Marylyn