JFK International Airport AirTrain

Taxi is your first option - yellow cab (from JFK to any point to Manhattan) costs flat rate of $45 + bridge toll($5.50)
There is also an express bus called New York Airport Service(NYAS) that leaves every 20 min in front of every terminal - it costs $15.
By public transportation:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Take subway E to 42nd street/Times Square stop.Cost of the ride: $7.25 per person.
JFK_E

Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Take subway E to Lexington Ave stop.
At Lexington Ave stop transfer to subway line 6 going uptown to 110th street stop. Your address is a block away on East 110th street between 3rd and 2nd Avenues.
JFK_E

It is very hard to find "cheapest and quickest" way - the cheapest way is usually the slowest and the quickest way is usually the most expensive.
- the cheapest way
Take Airtrain to Jamaica Station then take either subway
E or LIRR to Manhattan
- the quickest way:
Take Yellow cab. Price - flat of $50.50(for the whole taxi).
JFK_E

Hi John,
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Take subway E to Lexington Ave stop.
At Lexington Ave stop transfer to subway line 6 going downtown for one stop to Grand Central. Then transfer to bus M42 going toward United Nations. Get off at the last stop - your hotel is located right next to United Nations building on 1st Ave.
Note that the transfer from subway to the bus is free with the same metro card. Cost of the ride - $7.25 per person.

JFK_E

Inexpensive way would be mass transit:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then either:
-take subway line E to Penn Station or
-take LIRR to Penn Station(faster but more expensive)
Then take Amtrak to Penn Station in Newark.

More expensive but comfortable and fast way is a private car(that will accommodate 5 people) or private van(if you have a lot of luggage). It will cost you $100+ though(versus ~$50 for all five on mass transit). For that option go to Ground Transportation Counter(Welcome Center) and speak to the agents there.
JFK_E

You can either use the Airtrain or walk the distance. In either case it will take you around 5-10 min to get from T3 to T4. The Airtrain station at T3 is located across the street from the terminal.
JFK_E