Everyone posting here could give you a different restaurant recommendation and all be different - thats what happens when you have to feed 8M people.
That being said, this is what I do when I return....
After college I moved to CT and later Beacon NY when I was with IBM. During the non-summer months a bunch of us would head down at least 1 weekend a month and spend the day.
I recommend you DO NOT stay in the city at all and here is why:
goto Beacon, Fishkill or Newburgh NY and stay in the fine (cheaper - much cheaper) hotel of your choice. I myself prefer the quality or residence in Fishkill.
Saturday morning, goto the beacon train station - it is the largest in the Metro north Hudson line and the parking is mostly free on weekends (some inner spots remain permit only)
The train runs every hour at the same time and you buy your tix online or at the kiosk - NOT on the train. The trip takes 90 minutes and the scenery down the Hudson makes it worthwhile. The tunnels by peekskill, west point, garrison, croton harmon, sing-sing, all the 'fishing' villiages, spuytin duyvil, harlem river then the dive underground on the park ave viaduct.
WAY WAY worth the ticket!
The train arrives on the main floor just outside the grad concouse of GCS. when you get in, look UP and try to locate the 2 bricks not cleaned when they restored the place using simple green for the ad campaign.
Its lunch time right? either take S or 7 or walk (I recommend walking) from GCS to times square.
as you head north on broadway, at 211 west 43rd, off to the right down the street is the 'times square deli'. Its my fav but like I said, you will get 4000 recomendations.
While in times square, if you want to see a show, on SAT and sunday goto the TKTS kiosk and see what they have for sale. A lot at half price, some at only 20-30% off. The sale show is usually the matinee.
AVOID eating on 'restaurant row' (loose collection of restaurants in the 45-47 street range) overpriced pretentious slop.
Museums? When I was at IBM we got free admission cuz of the money IBM gave each year. Now I am a member of carnegie and I get reciprocal rights to the Met, but not the AMoNH. For first timers I recommend either. For younger kids who like monsters the AMoNH, for teen girls, the Met.
To get around I recommend learning how to subway! get a day card and plan your moves to make it pay. Last time I was there the day cards did not allow buddy swipes - once swiped it had to sit for 20 minutes. Take the Lex (green) express up to 86 for the met, take the B or C up to 81st.
On days where we did BOTH museums we would ride up the lex to 86 and the met, then walk across the great lawn to the AMNH and later B/C train down to 59th (columbus circle) and walk broadway down to times square.
Now for dinner, My fav is italian. if you dont want little italy, 134 W46th is la strada - it was good, and while technically in restaurant row, it was outside the beaten path.
but my REAL fav is Il Fornaio on mulberry. (lex down to canal, go thru chinatown to mulberry, turn left and walk, its on your right). not to far away is Puglias (featured in 'Big Daddy') but the food was not that great, but Jorge Buccio is there...dunno if he still does styx tunes
the staten island ferry is indeed free - but they seem to crash a lot of them!
for a return trip home, the hudson line runs every hour on the hour for the express (first stop marble hill or thereabouts) you would want this train as the others to poughkeepsie (which stops in beacon) are either locals or change trains in croton harmon.
so thats my 50 cents worth...