The double decker bus tour is an absolute must.
My husband and I lived in Manhattan for 3 years, over 20 years ago, but a few years ago, when we made our first visit back to show the kids the city we love so much, we decided to do the bus tour, even though we really do know the city well.
And were we pleasantly surprised. The bus tour is excellent. The guides on all the buses we were on were very knowledgeable. And the view from the top of the bus can't be beat.
I would take all the loops (I think there are 3 of them now) if I were you.
When we go back this summer for what has become an annual 3 day visit, we'll do all 3 loops (so far we've only managed the south loop).
As for other things, it depends. The first year we went, the kids were about 12 and 15 and both were terribly overwhelmed by the size and masses of people. They initially hated it, so I decided to take them on the movie tour of new york. Not anything organized. We took them to sites that were in movies (e.g., the steps of the library and central park west that are both in ghostbusters, grand central station from any number of movies, etc.etc.). It was fun and good for that age. And it got them 'in the mood' for the city.
As for must-sees, I can only tell you what ours are: the site of the world trade center (my husband worked there when we lived in Manhattan), Rockefeller Center (where I worked), Central Park (the zoo and then just wandering around wherever our feet take us - we used to live across from central park, so it's like coming home for us), the Museum of Natural History (on central park west), the Metropolitan Museum of Art (on 5th avenue), FAO schwartz (on 5th avenue), a walk from Lincoln Center up Columbus and then over to Broadway, up Broadway to Zabars (that's a long walk, if you're not into walking - which we are), walking all over the midtown area (park, 5th, 6th, the herald square area and up to central park, walking around Greenwich Village and SoHo.
The point is, New York is a town to walk in. If you like to walk, this is the place.
As for the subway - nope. It may have improved. I've heard it's better. But we were way too traumatized when we lived there to ever set foot on the subway there again. When we visit, we walk and take taxis.
It's true that TKTS is the place for cheap tickets, but since we're there for only 3 days, I always buy our tickets in advance at full price. I want to make sure I have good seats in the shows I want to see (this year I have tickets for Avenue Q and Chicago).
We stay at the Waldorf Astoria (on Park Avenue), which is much more reasonable than you might think. About half the price of similarly classed hotels.
We stay in concierge, since the cost is recovered in the breakfast, snacks and drinks that we get in the concierge lounge.
Oh, and of course you should also take the ferry to the statue of liberty and ellis island. Lots of history there. We did this a few years back, 2 weeks before 9/11. The view from the ferry of the world trade center was absolutely stunning, but i don't know what the view would be now.
If you suffer at all from seasickness, I would avoid this ferry ride. We did it in the morning and both my daughter and I were finished for the rest of the day. We had to go back to the hotel to lie down. We had planned on going up to windows on the world, but we were so sick i said, it's ok, we'll go next year. It's not going anywhere. Little did I know that 2 weeks later it would be gone. I hope they re-build the towers one day. They were magnificent.
There's a famous ice cream parlor on central park south that's really nice, but my senility has wiped its name from my memory.
Oh, and one more thing that you should try - New York Italian Ices. You can buy them on the street from the "umbrella cart" verdors. Great stuff.
I had that every single day for lunch when i was at work. yummy.
There's tons of other stuff to do, but that's all that we have time for.