Hi! I live in one of the boroughs of NY and work in the City.
The first thing I would suggest is determining a budget. You can do NYC on any kind of budget but that will play a big factor in your stay and what you decide to do, eat, Broadway or off-off Bdwy shows, etc.
If you want to see a show(s) start planning what you may want to see. Very popular shows sell out months in advance and if you know what you want to see, buy the tickets. Make a loose plan of what you want to do each day so you are not hopping from downtown to uptown and back downtown again. The Broadway shows are midtown. Being the proud mom of a Marine (please thank you son from the bottom of my families heart

) I am sure you want to visit Ground Zero, although it looks nothing like it did the months following 9/11 (thank goodness). Ground Zero is downtown. Museums and the high-end boutiques, shops are mostly found up/midtown. Carriage ride is upper midtown and you can do that through Central Park. SoHo and the Village are not downtown but not midtown either. These are places you also want to visit, walk around, browse. You can have a big meal or sit in/outside a cafe and relax and people watch. Love Italian, Little Italy is not to be missed. As someone mentioned, Yankee Stadium is in the Bronx not in the city. I'm not sure if you could "visit" if there wasn't a game going on. There wouldn't be anything going on if there wasn't a game. Try Shea too

(We're Met fans LOL)
Where to stay will depend on what kind of atmosphere you want. Downtown, Gramacey Park, Midtown, Uptown all have something to offer. I do like the great restaurants that are offered in the Gramacey Park area, which is also residential. Maybe determining what you will be doing each night (ending with a show, restaurant or club) should determine where you want to stay. Enjoy and have fun!!