Yes, you can pick up a cab at the airport or at the hotel, no need to make pre-arrangements.
At the airport, when you exit the terminal there will be a taxi stand with a line of yellow cabs (and sometimes a line of people waiting for them, but it usually moves quickly). From
JFK there is a flat rate of $35 plus toll and tip (it is customary to tip cab drivers here), so you don't have to worry about getting ripped off. This is usually posted all over the inside of the cab.
When you leave your hotel, you can just hail a cab anywhere on the street (unlike some cities where you have to find a taxi stand). At the bigger hotels, there may be cabs lined up waiting for people, or you can ask the doorman/bellman to hail a cab for you (give him $1 if he does this). The cabbie will start the meter when you get in the cab, so there is no need to negotiate a rate. Although I suppose it is possible for the cabbie to take a circuitous route, from my many years of experience NY cab drivers are generally very honest and would rather get you to your destination quickly so they can pick up another passenger. Again, the toll will be added separately.
For tolls, the driver might ask you to give him the cash ($4.00) when you reach the bridge or tunnel, or if the driver has E-ZPass (an electronic toll payment system) he will just add $3.50 to the total amount.
As for tips, it is totally up to you and should depend on the service you receive, but as I mentioned above, it is customary (unless of course the driver is totally rude or whatever) to tip about 10-15% of the fare (then round to the nearest dollar). So, on the drive in from
JFK to Manhattan, you can expect to pay about $43-45 including tip and toll.
Common sense is not so common. -Voltaire