A hundred or more years ago, three different subway lines were built. They are called the IRT, the IND, and the BMT. That is why some subway routes are named with letters, others with numbers. Today the distinction is not important, as there are many interconnecting passageways from one to the other. These are usually underground and out of the weather. All use the same payment scheme.
A rather wordy but good introduction is here.
Here is another good introduction.
The subway stations and platforms can be very crowded. At many stations, there are interconnecting paths to several other lines. Look for signs and read them carefully. You will be surrounded by swarms of people criss-crossing around you who have done this a thousand times and know exactly what they are doing.
You should be aware of express trains, as their use saves a lot of time and tedium. On the subway map, stations with express train stops are white circles; those with local only are black dots. (Of course locals also stop at the white circle stations.) All of the stops I recommend using have express trains, except the A at Howard Beach and the 7 at Court Square. If you come from JFK on the A-train from Howard Beach, you should switch to an express at the earliest possible station, which is Euclid Avenue.
Typically, there are no express trains on Sundays or holidays, or sometimes late at night.
In a station with one platform for all uptown trains and the another for all downtown trains, the local trains will stop against the wall and the express trains will stop in the middle of the station. Express trains and local trains run on separate parallel tracks. Apparently the NYC subway is the only urban commuter transit system to have a 4-track layout with separate local and express tracks. So the express are "inside" and the local are "outside".
===== < < < local < < < =======
--- Platform where you stand ---
===== < < < express < < < =====
===== > > > express > > > =====
--- Platform where you stand ---
===== > > > local > > > =======