What should I do/know when I arrive at the airport?
You'll have two basic options to get from the airport to where you're going: uber/taxi (more expensive but faster) or public transportation (less expensive). There are a few options "between" those two, like the Newark airport shuttle, but for your first time I recommend either public transport or uber/taxi. If you plan to take public transport, download the "CityMapper" app and use their directions. It will be $2.75 to get where your going because you get free transfers between bus and train, but note that at JFK you have to use the airtrain which is a separate $7.75 fee. If you take uber or taxi, make sure to say no to anyone who approaches you as your leaving and offers to take you. They'll be fairly deceptive and charge quite a bit. Either order your uber and follow the instructions, or follow the signs to the taxi stand.
What's different from other countries/cities?
If you're just visiting, public transport (subway and bus) will be able to get you almost anywhere you want to go. And it's incredibly dense, especially in Manhattan, so plan on doing lots of walking. But the other boroughs (Brooklyn, Queens, Bronx) have a ton of great restaurants and culture, so don't just stick in Manhattan. Also, New Yorkers are actually very friendly! If you need help, don't be afraid to ask. Try to find someone walking with a purpose and probably headphones in and ask them for help, since they're probably actual New Yorkers that will know how to help.
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Where should I stay / live?
As long as you're near a subway stop, you'll be able to get anywhere you want with fairly minimal hassle. Hotels and real estate in Manhattan, especially downtown, will generally be the most expensive and go down in price as you move away from downtown Manhattan. Otherwise it's really up to you in terms of what you'll prioritize.
How to go from A to B?
Use the CityMapper app to see all your options!
Coffee shops / Coworking to work from
Most of the usual suspects here, Starbucks, WeWork, even McDonalds. There are a few unique ones like https://www.kettlespace.com/ but for short-term your best bet is probably https://www.regus.com
Unique things I can only do/buy here?
Too many to list, and they're probably all well known at this point. The classic insider tip is to not pay for a Statue of Liberty tour, but take the free Staten Island ferry which goes right by it instead. Otherwise make sure you try a $1 cheese slice at Two Bros pizza, eat lots of different types of food (there's too many great unique places to waste time at a chain), and try to venture out of Manhattan.
The must try restaurants?
Again, too many to list here. But I'd use https://ny.eater.com/ to discover some great places based on the type of food you like.
Where can I go for a weekend gateaway?
You're already in NYC! But if you must getaway, the Amtrak train will take out of the city quickly, or you can take the LIRR to the world famous Hamptons. My favorite is probably to rent an AirBnB somewhere in the Hudson Valley to get into nature without too much effort.