Despite all the poor lost tourist souls you see with their maps wandering all over the city in complete confusion, it actually is not THAT hard to get around Manhattan. Sure it takes practice (New York wouldn’t make anything easy for you, would it?), but once you get the basic info down it does make a lot of sense. A lot of it is memorization, and a good sense of direction doesn’t hurt. The latter you may not be in control of though.
The Grid
Not the Grid
Broadway
Highways
The Grid
From 14th street until around 110th street, the city did you a favor and made it very easy to find your way around. After 14th, it can turn you into a crazy person. It’s slightly easier above 110th, but gets more complicated the more you go up. Technically, some of the grid system even extends into the Bronx. But we will focus on what is commonly thought of as the main part of the grid.
The numbered streets run east to west (i.e.- 57th street, 32nd street). The avenues (i.e.- 7th ave, Amsterdam ave) run from north to south. From west to east the avenues are in this order:
12th avenue/Riverside Dr., 11th avenue/West End, 10th avenue/Amsterdam ave, 9th avenue/Columbus/Morningside Dr, 8th avenue/Central Park West/Frederick Douglass, 7th avenue/Adam Clayton Powell, 6th avenue aka Avenue of the Americas/Lenox Ave aka Malcolm X, 5th avenue, Madison, Park, Lexington, 3rd ave, 2nd ave, 1st ave, York ave, East End.
Not all these avenues are present throughout the entire stretch of 14th street on up, and there are other streets in Upper Manhattan that are not listed here, but this is a basic idea of the order of the avenues. You can find below the exact order of the streets and their names, depending on where you are.
From 14th street to 23rd street (East to West):
FDR Drive, 1st, 2nd, 3rd, Lexington, Park Ave. South, Madison, Broadway, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th/West Side Highway
From 23rd street to 34th street (East to West):
FDR Drive, 1st, 2nd, 3rd, Lexington, Park Ave., Madison, 5th, Broadway, 6th, 7th, 8th, 9th, 10th, 11th, 12th/West Side Highway
From 34th street to 42nd street (East to West):
FDR Drive, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, 6th, Broadway, 7th, 8th, 9th, 10th, 11th, 12th/West Side Highway
From 42nd street to 59th street (East to West):
FDR Drive, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, 6th, 7th, Broadway, 8th, 9th, 10th, 11th, 12th/West Side Highway
**Sutton Place begins at 54th street one crosstown block east of 1st ave.
From 59th street (aka Central Park South) to 66th street (East to West):
FDR Drive, York, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, Central Park West, Broadway, Columbus, Amsterdam, West End, West Side Highway
From 66th street to 72nd street (East to West):
FDR Drive, York, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, Central Park West, Columbus, Broadway, Amsterdam, West End, West Side Highway
From 72nd street to 79th street (East to West):
FDR Drive, York, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, Central Park West, Columbus, Amsterdam, Broadway, West End, West Side Highway
From 79th street to 110th street (East to West):
FDR Drive, East End*, York**, 1st, 2nd, 3rd, Lexington, Park, Madison, 5th, Central Park West, Manhattan Ave***, Columbus, Amsterdam, Broadway, West End****, West Side Highway
*East End ends at 90th street
**York ave. ends at 92nd street
**Manhattan ave. begins at 100th street
***West End ends at 107th street
Not the grid
From 110th street on up: My best advice is to look at a map. It gets seriously messy, with avenues changing names and crossing over each other and essentially running amok. 
From 14th street on down: Again, look at a map, because this is even messier. Because it is really not even close to being on the grid system, the best advice is to memorize directions to each individual place you need to go. Eventually, you will start to get a hang of which miniature street is next to which other miniature street. See below:
Broadway
You might notice in the list that Broadway is wandering all around Manhattan and skipping over random avenues. Broadway runs from the Upper tip of Manhattan to the Lower Manhattan, and it moves diagonally across the city. It moves from West to East as you go further downtown. To know where it is, the best trick is memorization. Luckily, it is easy to figure out where it is based on the grid system because it crosses avenues at relatively major cross streets (i.e.- 59th, 42nd, 34th, etc.)
Highways
There are two highways that run along the outer edges of the city. On the West side, you will find the West Side Highway (how appropriately named) aka 9A or the Joe DiMaggio Highway. On the East side, you will find the FDR Drive, which also less commonly called the East River Drive.