We want to stay in Midtown Manhatten. Some of the hotels are pricey though.
No getting around it - you will pay quite a bit for any decent hotel. Just accept that as part of the cost of the vacation. Empire State Building, Statue of Liberty are 2 mainstays of any sightseeing goals. But do your research about Statue of Liberty. Lines for the ferry and actual entrance into the statue are VERY long.
Also consider: 9/11 Memorial Museum and One World Trade Center in lower Manhattan. Also, take some long walks in lower Manhattan' that's the Wall Street area, and also the Battery which is next door at the southern tip of Manhattan. This part of Manhattan was where NYC first settled, and you can tell by the streets, many of which are not the typical right-angle gridwork.
Also in lower Manhattan is the Hudson River Promenade, nice place to walk along the Hudson. Speaking of walking, Central Park is another oasis to get away from the noises and hustle-bustle of the City. If you have some time to kill, you might want to take the Staten Island Ferry (it's free!).
Rockefeller Center is a popular site to visit, but I have to confess I'm not sure why. Most definitely, take in at least one Broadway play. Tickets can be expensive, but in my opinion it's worth it.
Depending on your dining tastes, dinners can be very expensive. But there are many moderately priced options. DW and I both eat simple meals; in that vein you may want to consider Junior's, a New York institution with GREAT cheesecake for your dessert. By NY standards Junior's is modestly priced.
Finally, the NYC subway system is a better means of transportation than bus or taxi. The subway map may look confusing at first, but trust me you will learn very quickly, and it's not as difficult as it may initially seem.