What is the best way to get from Newark International Airport (arriving from UK) to Mid-Town Manhatten (staying near Grand Central Station)

If not much luggage, the train would get you to Penn Station and you could walk or cab to hotel. You take the airport monorail/train to the stop where you pick up either Amtrak or NJTransit (cheaper) train onto Manhattan.
Cab would cost more but drop you off at the hotel. Depending on time of day there may be traffic delays.
There is also a bus service that runs from EWR to a drop off very near GCS.

Car service:

www.dial7.com

If you have little luggage and time to kill, then you could consider public transportation or a bus, but unless you're destitute, it's not worth it in my opinion.

Enjoy NYC!

I should have mentioned that a car service is the same price or less than a taxi.

How many of you are there? How much luggage do you have? Are you able/do you want to haul it all over yourself?

A car service or taxi to GCS will cost about $60. Shuttle buses are about $17 per person (check EWR site for details). Public transit (Air Train to regular New Jersey Transit to Penn Station - NY NOT NJ - and then cab or walk - almost a mile) will cost considerably less for 1 or 2 people if you don;t mind dealing with luggage yourself. (Cab from Penn Station to Grand Central will be no more than $10.)

Train is cheapest but doesn't get you near Grand Central without a lot of changing. There are Express Airport buses for about $15/person that make a stop right near Grand Central. Compare that to about $60 for a taxi or car service for up to 4 people with door to door service.

Take the airport bus service. One of them goes to Grand Central Station after stopping on 34th and 8th Ave.