New York City has a great public transportation system and you can use it to easily navigate almost all of the city. The two primary modes you will use are the subway and the bus, though the majority of places you can get to just with the subway.
Subway
The NYC subway is the largest rapid transit system in the world by number of stations (469) and is located in all boroughs of the city except Staten Island. There are 34 lines in operation, each with a letter or number and a corresponding color. There are several sets of lines that operate along the same path for part of their journeys and then branch off at some point to their individual routes (usually at points outside of the main Manhattan city center) and those will generally be color coded the same to show uniformity.
There are two kinds of subways – local and express. These often use the same platform, but local will make many more stops and the express will stop at just the major stations. If you get on local when you needed the express, you can always transfer to the express at the next major station. Conversely, if you get on the express when you needed the local, you can get off at the next stop and then transfer to the local in the opposite direction you were travelling if you’ve already passed your stop. These transfers are free, as are all transfers to other subway lines as long as you make the transfer underground and don’t exit the station.
The 1, 2, 3 (red) lines run North and South along the west side of Manhattan up to the Bronx and down into Brooklyn. 1 is the local line and 2, 3 are the express. These are the lines you will want for areas such as: Wall Street, World Trade Center, Penn Station, Times Square, Columbus circle, Lincoln Center, and Columbia University.
The 4, 5, 6 (green) lines run North and South along the east side of Manhattan, up to the Bronx and down into Brooklyn. 6 is the local line and 4, 5 are the express. These are the lines you will want for areas such as: Wall Street, Brooklyn Bridge, Union Square, and Grand Central. This and the 1, 2, 3, mentioned above are likely the two sets of subway lines you will use most.
You will probably also use the S (gray), which is the shuttle from Grand Central to Times Square, coming back and forth constantly and exclusively between those two stations.
There are subway lines that criss-cross around the city as well, but for most touristic purposes, the lines mentioned above will be sufficient. The main exception is the B, D, F, M lines (orange), which include stops such as Rockefeller Center, Bryant Park, and Washington Square. F and M are local trains, B and D are express (at least in the central areas where these lines converge).
To use the subway, you can purchase either an individual ticket, good for only one use, or an unlimited pass. Unlike some other subway systems around the world (London, DC), the subway fare for an individual ticket is fixed price irrespective of how far you travel or how many underground transfers you make. In fact, one could literally pay a single subway fare and ride every subway line around the city without having to pay more (note: this is only recommended for those with lots of spare time on their hands).
The main areas in Manhattan where subways do not reach effectively: the far west (west of 8 avenue), the very east (east of Lexington Avenue, particularly the Lower East Side), and traveling from the east side to the west side north of the beginning of Central Park (59th street). In these cases, your best bet is to find a bus route.
Bus
NYC buses are underused, but can be essential in those few areas where subways don’t do the trick. One thing to keep in mind, though, is that unlike subway routes which are quite easy to master, bus routes are not unless you frequently travel on a certain route. And unlike in the subway, there will be no map of bus routes to consult on the bus or at most bus stops, so you need to know where you’re going before you try to take a bus. For this reason, it will be incredibly useful to download a navigation app to your smartphone. Our preferred one is Citymapper and we rely on it in every city we go to (it is very useful for the subway as well)!
To ride a bus, you will need to purchase a ticket similar to the subway ticket above. You can do a single trip or purchase an unlimited pass. This can be used interchangeably on both the subway and the bus. If you need to transfer buses, then ask your bus driver for a free transfer ticket that you can then show on the next bus. If you are transferring from the bus to the subway or vice versa (within 2 hours of first swiping), then the free transfer is automatically registered on the card. No transfer ticket necessary.
MetroCard Pricing
$121 for a 30 day unlimited pass
$32 for a 7 day unlimited pass
$2.75 per ride if you buy multiple at a time, otherwise $3 for one ride only
$1 charge for a new MetroCard (don’t throw them away, you can refill the same one!)
Taxi 
The iconic yellow cabs are everywhere in the city and are incredibly easy to hail. Just stand at the curb and stick your arm out towards the street. During rush hour or rain, hailing a cab can take a while as cabs are all full. But this is likely a time when you do not want to take a cab anyway since traffic will be a monster and you will likely get everywhere faster by taking the subway (provided you are near a subway station – which you almost always are).
You can tell when a cab is free if just the center light is lit. If the side lights are lit, then the cab is off-duty (the side lights say off duty on them, but can be hard to read). If no lights are lit, then the cab has a passenger inside. We generally just hold our arms out regardless of what light is or isn’t on just in case. Most cabs accommodate up to 4 people, which can end up being cost effective if you’re a full car. There are some larger van cabs that can hold more.
When a cab stops for you, get in, sit down, close the door, and then let the cab driver know where you are going. Cab drivers are required to go where you ask them to, including places like Brooklyn, though some do not want to make this journey and will decline you if you ask before you get in. Why? Because it is less likely they will pick up a passenger on the way back from Brooklyn into Manhattan and they feel as though they are losing a fare. Don’t give them the choice.
As soon as you’re on your way, make sure the meter is on. Sometimes there are surcharges for certain times of day or tolls you will have to pay. Those will get added to the meter, but if you think you’ve been mischarged an extra, just ask about it. Remember, all cabs have an explanation of their pricing inside the cab itself as well as cab license numbers prominently displayed if you need to make a complaint or a report.
Almost every cab accepts credit cards and often it’s part of the TV screen in front of the back seat. You just select the “pay by card” option on the screen and swipe your card when you’re at your destination. You can always ask for a receipt.
One thing to pay attention to is that you will automatically be given 3 choices for tip: 15%, 20%, and 25%. These have been greatly inflated over the past few years, when the options used to be 10%, 15%, and 20%. You should feel perfectly happy giving 15% for good service, but can certainly give more if you’d like. There is also an “other” option where you can enter in exactly what you want. Or you can decline tipping completely if service has been unacceptable.