New York City has 3 main airports: John F Kennedy International Airport (JFK), LaGuardia Airport (LGA), and Newark Liberty International Airport (EWR). LaGuardia is the closest into Manhattan, JFK is the biggest international airport, and Newark is a convenient 3rd option just outside the city in New Jersey. Since New York is one of the busiest cities in the world, each of these airports have their lists of good and bad things, so it really doesn't make too much of a difference where you fly in to.
Even though New York has one of the greatest public transit systems in the world, they really don't have a great of a public transit system to/from the airport and into the city. The easiest way to get into the city from any of the airports is going to be a taxi.
From JFK, you can take a taxi for a flat rate of $52 plus tolls and tip and it can take anywhere from 35-90 minutes depending on traffic and construction (this really is my least favorite airport option...). You can also take The AirTrain ($5) to the MTA (Metropolitan Transport Authority) subway, which will cost $2.50. This trip will take a little over an hour into the city. There are also express buses into the city (to Grand Central or Port Authority) that cost about $16.
LaGuardia is the closest airport into the city, but the least accessible by public transit (this airport is my personal preference of the 3 because of proximity to Manhattan). The best option is taking a cab that usually costs about $30-50 plus tolls and tip depending on traffic and where you are going. The total drive time usually takes between 20-40 minutes. There are a few bus options that will cost about $2.50 for a one-way ticket, but it can take about an hour and a half to get into the city.
A taxi from Newark will take about 45 minutes and cost about $60-80 plus tip/tolls, making this the most expensive option via taxi. There is also a $17.50 surcharge on top of the normal taxi rate if you are heading to the airport from the city (I literally had a $100 fare to Newark when I had to take a cab there one time). You can take The AirTrain from the airport and board any train bound for New York's Penn Station for $12.50. This trip will take about 30 minutes assuming no delays. Similar to the other airports, there are also bus options (costing about $16) if you want to look into that too.
New York is the motherland of taxis. There are taxis virtually everywhere in this city (except during rush hour and the afternoon times when they are switching shifts...ahhh!) and they are not hard to hail. Car shares like Uber are also popular here, so feel free to use that app as well.
A few years ago, the Bloomberg administration launched Citi Bike, which is the largest bike sharing program in the US. For about $10 per day, you can rent one of these bikes at any of the kiosks located around the city. It is a great way to get around when it is nice out if you're into biking in the city.
To get to some of the other boroughs of New York from Manhattan (or just to catch some amazing views), you can access the city's water taxis or ferries. The East River Ferry connects to various locations in Manhattan, Queens, and Brooklyn. New York Water Taxi ($30 for an all-day pass) has a great service around Manhattan and Brooklyn, and the Staten Island Ferry is a free way to get across the New York Harbor.
New York's MTA Subway and Bus systems are a pretty easy and reliable way (and cost effective) to get around the city. Fares start at $2.50 (or $2.75 for bus) and include uptown, downtown, and crosstown services. You can purchase a MetroCard to use starting at a minimum total of $5.50 at station booths or $10 at the vending machines in the stations.
Make sure you pay attention to which direction the subway is going before you walk down into the station! Lots of people make the mistake of walking down into an uptown station when they should be heading into the downtown station (which is most likely located across the street) or vice versa. Use Google Maps or other similar apps to find the best routes to take.
Walk!! This city is the greatest city explored by walking!
