The answer obviously depends on where you are flying from and what airlines you like to use (for example, American generally flies to LaGuardia; Continental to Newark; JetBlue to JFK). If you have the choice, LaGuardia is the most convenient -- it is the smallest airport and closest to Manhattan (about 25 minutes by taxi).
JFK and Newark are roughly equivalent in terms of convenience -- Newark is probably 10 minutes closer (although this all depends on the traffic) but is often an extra $10 in taxi fare. Expect a 45 minute taxi ride from Newark; close to an hour from JFK.
How to Get Into Manhattan from the Airport:
Frommer's offers a useful summary here.
TAXI
Approximate fares:
Newark -- $55 + tip
JFK -- $45 + tip
LaGuardia -- $25 + tip
AIR TRAIN
From Newark. The Air Train offers connecting service to NJ Transit and Amtrak trains which will bring you into Manhattan.... and often much quicker than taking a cab or bus. $11.55 one way will bring you from Newark Airport to Penn Station. You do have to change trains once in Newark and then catch a taxi at Penn Station so it is not an effortless system.
From JFK. The Air Train offers connecting service to two New York City subway stations in Queens. Unless you are staying downtown (in which case you probably don't need these instructions) you should take the Air Train to Jamaica Center and then get on the E. The total trip costs $7 and takes about an hour. Again, this journey requires at least one change of trains plus navigation of the NYC subway system... so it may not be for everyone.
CAR SERVICE
Catherine recommends Mirage Limousine at 718.937.6600 or (800)4-MIRAGE.
BUS
There are several bus services at each airport that take you to midtown for about $15.
Getting around Manhattan
TAXI
If the middle light is on, the cab is available.
SUBWAY
Mingle with the people. [click for subway map]
