Here's the phone/internet situation situation in the Caymans.... It's not pretty. The island is controlled by the Cable and Wireless monopoly. Thus, services are very expensive. If you want to make a call from the Caymans, make sure to buy an ATT phone card in the states before you leave - with lots of minutes. You will get approx 1 min for every 14 on your phone card. Still, not overly expensive, considering how cheap you can get a phone card.
As far as the internet..... There's about 4-5 places in Cayman with open internet access...it is high speed. Cost is .25-.50 a minute. The best place to use the net is in a place called PD's Pub, in the Seven Mile Shops, right on 7 mile beach.
Hope this helps, Let me know if you need anything else.