I just returned from the Zenith a few weeks ago. There are several locations where the internet is available including the show lounge and library and I actually never saw anyone using them. I didn't use the internet onboard so I'm not sure how quick it is. I can't remember exactly how much it cost but I believe it was somewhere around 25-50 cents per minute. I hope I helped. Happy cruising!
RCCL's Majesty of the Seas * Eastern Carribbean - '93
Celebrity's Horizon * Bermuda - '95
Celebrity's Zenith * Bermuda - '98
Celebrity's Horizon * Bermuda '99
Celebrity's Century * Western Carribbean '01
Celebrity's Zenith *Bermuda - '04
These computers are connected to the internet by satellite, so they can be slow, but are useable to check email, but pretty expensive to do any surfing, etc. The charges rack up pretty quickly, a few dollars per session for me to check email every day.
There are some free sites you can access on board (like celebrity's site), but make sure you do those before swiping your room card (which looks like a credit card) or the meter will be running.
It seemed like the connection was quicker in port when the ship wasn't moving. Also - be sure to completely log out after you've completed your session so it quits charging you. The final screen you get (at least on the Zenith) showed your charge to your room for the session. I always logged out twice just to make sure.
In March on the Infinity you could send an e-mail for $2.00 and receive one for $2.00. I sent the e-mail to my son and he forwarded them to other members of the family and the total charge was only $2.00 and you could have the computer as long as you needed to write it. This was much cheaper than getting on the internet. They also would let you look at your digital pictures free and had a reader you could use.