Here's all the schedules:
mta.info | MTA Schedules
The subway lines and a lot of the bus lines run frequently enough that you don't have to time yourself for a specific train/bus. This is especially true if you're going to spend most of your time in the touristy areas.
Here are the different maps (borough bus maps, subway maps, and the commuter rail maps):
mta.info | MTA Maps
And yeah, I think they have a "Hudson News Stand" that sells MetroCards in the airport.
You have two choices: Pay-Per-Ride (PPR) and Unlimited. If you buy a PPR MetroCard with a value of over $10 (or refill in an increment of over $10), you get a 7% bonus, which reduces the fare you pay from $2.25 to $2.10.
A weekly pass is $29 and a monthly pass is $104. They used to have a daily pass, but they discontinued it. You need to ride 14 times on the weekly and 50 times on the monthly to come out ahead (or else, you're better off with Pay-Per-Ride)