The Internet is .50 per minute. If you are used to anything above dial-up, you will find it to be tediously slow. I have a comcast e-mail account, and since their pages take a bit of time to load, I found it was costing me at least $4 just to send a 'quick' e-mail. Next time I would use a quicker webmail, such as hotmail.
For my last cruise, I researched Internet Cafes ahead of time for the ports we visited. In Jamaica, I paid $4 for 15 minutes of access, and although the computer was quite old, the Internet was truly high-speed.
As far as cell phones, it will depend on your service and your ports. I was able to get a signal in most ports, and at sea while close to some land, but with my current plan, long distance and international is expensive, so I did not use it.