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I would definitely recommend baby wearing if you are planning to take transit! But I would also bring a lightweight umbrella stroller if your baby is heavy since there is a lot of walking if you want to sight see.

I loved visiting the Met and the Guggenheim. I also really liked walking through Central Park. Times Square is interesting but it's all tourists. Still interesting to see once though! I also loved seeing St Patricks cathedral, the New York public library (inside and out!) and Rock plaza which are all near each other in midtown. On a clear day you can see the Statue of Liberty from the Wall Street area. And I would definitely cross the Brooklyn bridge on foot. I loved that! It's busy with lots of cyclists. You can walk one way and subway back.

If you're into macarons, Laduree is near the Guggenheim on the UES. Levain bakery has delicious chocolate Chip cookies on the UWS. I really liked brunch at Beauty & Essex in lower manhattan.

Is central time 1 hr difference from eastern? If so, I'd just keep my baby on the same schedule! I've done that before and it was great.

What time do you arrive? I am always antsy when I arrive in a new city at night so I would just take a cab. But if you a arrive early you could attempt to take the subway.

Depending on which airport you arrive into, you may have to take a cab. If you don't want to bring the car seat, you can try a car service and see if they will provide one. Once you're in Manhattan, you won't need the car seat, it's just getting to/from the airport that's annoying. We just flew to Florida and used our stroller with the car seat adapters, is that an option?
Places I'd recommend- The Highline, Children's Museum of Manhattan, walking the Brooklyn Bridge, shopping in Soho...
I would just use the hotel crib and bring your own sheet, but that's me, some might prefer not to.
I looooove love love the brunch at Barking Dog, which is right by where you're staying. They have the best biscuits!

I second all three food recommendations from @sunny. Also, if you like sushi, you are near my absolute favorite sushi restaurant - Sushi Sen-Nin, on 33rd St between Park and Madison. It's a bit pricey but totally worth it, even if you just do takeout.

If you are flying JetBlue, you're most likely going to go into JFK. It's possible to take the AirTrain from JFK to the Long Island Railroad, directly into Penn Station. It's definitely a bit of a hike, but that would eliminate the need for a car seat.

Sounds like a great trip!

-Any tips on keeping a 9 month old happy on a plane?
We have had luck with lots of snacks and using random objects as new and exciting toys - hits include plastic cups from the flight attendants, styrofoam cups, coffee stirs, things like that.

-Tips on adjusting for the time change? (we're coming from Central time zone)
Since it's only an hour difference, I wouldn't adjust at all. Just keep your same schedule from home and shift forward an hour.

-Is bringing a car seat for the cab something I should consider, or should I try and stick to public transportation?
Are you flying in to LGA or JFK? As it's your DH's first trip to NY and your first vacation with your LO I would strongly recommend taking a taxi from the airport, and you should use a carseat if you do. Is your LO still in an infant carseat? Do you have a carseat attachment on your stroller? If so that would be the easiest way to go - then you can wheel the carseat around the airport, and leave it in the hotel if you choose to walk / use public transport when you're in the city.

-Bring a PnP or use hotel crib?
I'd call the hotel and ask if they have cribs to use, and ask what they are - hotels I've seen have had their own PnPs that they use as cribs, and they're fine.

-Umbrella stroller or carrier? Both?
Definitely bring a stroller if you're going to be doing a lot of walking. A carrier might be useful if you love using a carrier, but it's probably not necessary if you and your DH can lift the stroller up and down subway stairs.

-Any restaurant recommendations in the area we are staying in?
I'm not familiar with many restaurants in that area, though this is a fantastic website for restaurant reviews and recommendations, which you can browse by neighborhood and price point: http://nymag.com/restaurants/

You are right on the 6 subway line, though, which is fantastic - I'd suggest exploring along that line. Union Square/14th Street is a couple stops downtown (and within walking distance) - they have a fun farmer's market in the park each weekend; Astor Place is one more stop down and you can see Washington Square Park and walk around Greenwich Village from there; go one more stop downtown to Bleecker St. station and wander around the little boutiques and great cafes in Soho and Nolita. (You can tell I'm personally partial to downtown Manhattan, but previous posters' recommendations for uptown places are great too!)

Let me know if you need any other tips or ideas.

How exciting! I live in NYC and just returned from a trip to Florida yesterday with our 9 month old, so I'm going to try to combine both experiences to give you some tips!

In no particular order...

1. I know it's a lot of stuff but I would probably bring a car seat, umbrella stroller, and ergo. My logic is this: when we tried to get a car service with a car seat to take us to the airport, the cheapest estimate I could find was $130 for a one way trip! Yikes! And, we literally live in Queens, only about 15 minutes from Laguardia so it's not as though it's a long drive. With that in mind, we decided to take the car seat and the airline didn't charge us for it. It's a lot to carry but I didn't feel comfortable taking a cab without one and I also didn't want to pay $260. Plus, I figured it was easier to lug a car seat than to try to get all our bags around via public transportation.

If you want to rent a stroller while here to cut back on your stuff, here is a link I found with some local rental spots:
http://mommypoppins.com/newyorkcitykids/nyc-stroller-rental-car-seats-cribs-baby-equipment-gear

Our LO happens to love the Ergo and that's his preferred mode of travel so that's why I would need it. But, if yours is a happy traveler and napper in the stroller then you can probably leave it at home.

2. Be sure to bring plenty of snacks on the plane! Our flight was delayed by one hour once we were already on board and we were able to keep LO entertained for a bit of that by feeding him puffs and MumMums. Random little toys and rattles definitely helped as well!

3. When in NYC, my favorite thing to do is just explore new and interesting neighborhoods. Personal favorites are Williamsburg, Chinatown, the Lower East Side, and the West Village. If the weather cooperates, there is nothing lovelier than a stroll in Central Park on a warm day!

4. I love the Guggenheim and the Met (I work across the street!) but if you want some museum experiences other than art, the Tenement Museum, the Intrepid, and the Transit Museum (in Brooklyn) are all amazing as well. Note, the Tenement Museum is only available through tours that must be reserved in advance.

5. For classic NYC skyline photos, walk over the Brooklyn Bridge to the Promenade in Brooklyn Heights. There is a beautiful carousel in the park there that might be fun too.

I see there is already great advice given by other Bees. Have so much fun!!!

@sunny: we attended Easter mass at ST. Pat's and it was fantastic! The hats and everything were so fun.

@Mrs. Sketchbook: I used to attend mass there when I was in college/living in the city and completely took it for granted. It's a beautiful place.