If weather permits a great way to see New York without spendng a fortune and getting a dose of daily life is biking. I will stress that if you want to do bike on the streets be very careful. Look out for parked cars with people in them as doors opening are the death of most bikers. Stay away from high traffic areas (midtown) as best you can because all New Yorkers in cars, or walking will torture bikers who don't know how to work the streets. I'm a New York bike rider, and I avod midtown during the 5 - 7 pm slot.
There are two great places where you can catch more views of the city and more attractions then most realize. The first is West Way. Former Mayor Gulianni spent a great deal of money and a time creating a walking, running, biking path that runs along the entire west side of Manhattan (along the river) and even around lower Manhattan to the east side. This path is only for runners, walkers, bikers and roller bladers so you're safe and can be lesiurely with your riding. Start down at Battery Park if you can. Then ride North. You will over the course of your ride see the Statue of Liberty, Battery Park memorials, the Jewish Museum, the Downtown West side Yacht Club, World Financial Center, New Jersey views, and that's just the first mile. When you hit the World Financial Center depending on the time, lock your bike and sit at one of the outdoor cafes with some brunch while taking in the view. Often during the spring and summer there are events happening on the weekends you can take part of. If you Continue to ride up the path you'll pass Chelsea Peirs (you can stop here and play a round of golf), some great old structures of the MTA, the Circle Line, The Intrepid (and if you're a far biker) you can see the 72nd street Marina which is always cool given Manhattan is not known for its houseboat community which is located here.
If you want an escort, check out the central park tours, they will provide you with one and a bike (good for kids) so that you can ride through Central and know exactly what you are seeing. If you choose to go Central Park alone, veer from the "loop" (the main road) and take the off the beaten path roads so that you can discover hidden areas of the park like the life size chess area, the hedge garden, the zoo (no one goes to the zoo it's amazing, and free sea lion shows). You can also find the boathouse and row around the park. Check out the roller blading area where people are disco dancing with wheels. The park is a bevy of activity and around every corner you'll find something to get off your bike and participate in.