New York City
Make sure you get an MTA card for the subways. You can get one in any station in the city from a Kiosk. Only takes credit cards. $30 for a week should be enough, and you can always reload.
Use Uber if you get lost or you're tired. Taxis are hard to come by some days and the cost for UberX and a taxi is about the same. But don't trust the driver to know where he is going, especially in Brooklyn. Always, always have Google Maps on your phone to provide navigation to the driver and follow along. Many of these guys don't know the City and don't speak English that well. When you tell the driver where you're going, do it like this: "Heading to the Metropolitan Museum of Art on the Upper East Side at 5th Avenue and East 82nd Street." The place. The neighborhood. The street for the place address. The cross street. First, this approach will always get you where you're going. Second, you'll seem much less like a tourist. If you use taxis, that is for sure how you should do it.
To get around the city, Google Maps is your friend. So is City Mapper.
JFK to City
Use Uber to get from JFK to the city. Airtrans will get you to the A train and to the city, but that's a pain in the ass.
Take Uber from your hotel in Midtown to the Metropolitan Museum of Art on the Upper East Side at 5th Avenue and East 82nd Street. You don't want to try the train on your first morning. Orient yourself! When done at the Met, there should be a hot dog stand on 5th Avenue. Have one. Then, walk west through Central Park to the Museum of Natural History on the Upper West Side at Central Park West and West 79th Street. The walk is about 8/10 of a mile if you don't dawdle and get lost in the park. That said, get lost in the park. It's an amazing place.
When done with the museums, you'll be on the Upper West Side. There is subway station at the Museum of Natural History -- the B and C lines. From there to Columbus Circle Station is one stop on either line. The Shops at Columbus Circle are pretty cool and worth a walk inside if it's warm outside. The Mandarin Oriental Hotel is a few blocks away. Head up to the elevator to the Lobby Lounge for the best view of Central Park and a rest. This is one of my favorite places in the city. Order a soda or coffee or tea.
After a break at Mandarin Oriental, you'll be 8/10 of a mile north of Times Square. Walk south down Broadway to Times Square. Early evening is a great time to get there for the first time and the walk is fun. Once in Times Square, there are a gazillion touristy places for souvenirs and dinner. Have a bite and watch the lights come on. From Times Square, it's three city blocks east down 45th Street to your hotel. By the by, Ellen's Stardust Diner is fun. VERY Times Square with singing waiters and waitresses.
To do the touristy things in Lower Manhattan, head from your hotel to Grand Central Terminal (take time to look at Grand Central Market and the Great Hall). Also, the Chrysler Building is out the door at the end of Grand Central Market and the lobby is pretty cool. Anyway, take the 4 or 5 train to Battery Park Station. It's a 25 minute ride on a good day. Once you get off the train, make sure you walk over to see The Sphere. It is the sculpture the World Trade Center towers crushed on 9/11. If you want to see the Statue of LIberty, head south toward the water to catch a ferry to LIberty Island and the Statue. Note the lines to board the ferry are long and the security is tight. If you don't want to see the Statue, you can walk 7/10 of a mile to the World Trade Center. The 9/11 museum is worth a visit for sure, but the reflecting pools are pretty sobering. Not many tourists do it, but the FDNY memorial here is quieting: FDNY Ten House. It is the fire house closest to the site and the first responder on 9/11. When finished with all that, you can take the 4 express train (Lexington Avenue Express) back north to Grand Central from Fulton Street station or take an Uber. It's 4/10 of a mile from the memorial to the station for the 4/5. Alternatively, you could take an Uber to LIttle Italy for dinner at Pelligrono's. If you do dinner in Little Italy, you'll for sure want to take an Uber back to your hotel.
Take a stroll north on Madison Avenue (4th Avenue), west on whatever street you fancy (say 57th Street where Trump Tower is) and south on 5th Avenue back to your hotel on 45th Street. Lots of shopping to see. If that gets to you to afternoon, you'll need to get ready for lunch and a trip to Brooklyn. I'd take an Uber to Katz's Deli, have a late lunch, and then take another Uber to Prospect Park. That should finish your day. Get an Uber from Prospect Park back home.
Since you've spent a bunch of time in Manhattan, you should really head over to Brooklyn. Head back to Grand Central and take the F train to York Station. The station will leave you on Jay Street. Walk north on Jay Street three blocks to Plymouth and turn left. You'll find yourself under the Manhattan Bridge and then the Brooklyn Bridge and then in Brooklyn Bridge Park. This is probably the best place in the City for a photo and the Carousel area has had more movie and TV scenes than anywhere else. It's probably the most filmed and photographed area of the City.
For food, The River Cafe is a famous place for Sunday brunch. You'll need a reservation. If it ends up late enough, go to Juliana's for a slice or a whole pie. You could also try Grimalid's (super famous NY pizza). Then, stroll around DUMBO, which is the neighborhood you're in by walking back up Front Street to York Street and then get back on the F train. By the way DUMBO means Down Under the Manhattan Brooklyn Overpass.
If you have time and want to wander around SoHo and Little Italy when done in Brooklyn, hop off the F at Broadway-Lafayette St Station. Wander around the shopping of SoHo. If you didn't make dinner on Friday, this is the time to go to Pelligrono's. Same station. Whatever you do, get back on the F train back to Grand Central and get some sleep.
If You Have Time