This can get tricky. Officially MSC has email and Internet on all it’s ships. In reality it is not so clear.
WWW: On most MSC ships you have a white-listed Internet. You can get to your bank or mortgage company, for example, but high bandwidth sites like Facebook or skype is blocked. Most MSC uses commercial satellites and they do this to keep costs down.
On the AKE class you have unblocked Internet. The problem is that the AKEs piggyback off the USN birds so USN controls the bandwidth. On deployment a total Internet and email outage lasting weeks is common. The Arabian area (with lots of USN ships) is notorious for AKEs going dark.
Email: You must use your msc.navy.mil (or similar) account to receive email. You will not be able to access your personal email from the ship. Email attachments are limited in size. Usually this is 1 or 2 MB. So a few photos are ok, but a video will bounce. Again, as long as you are not on an AKE your email will almost always work.
Telephone: Some ships have a single public line to make calling card phone calls with. Many do not. In an emergency you can make calls threw the radio room with the captains permission.
Because off-ship comms is dependent on so many factors (ship class, captain, location, number of USN in the area) I personally do not count on it. I own an international cell phone (Verizon does not work well overseas), I have all bills set on auto-pay with my bank, and my family knows the only guaranteed way to reach me is with text messages and snail-mail. (I know many people roll their eyes at snail-mail but a hand written letter is priceless.)
More and more people are buying sat-phones for personal use. It used to be only officers but these days unlicensed will bring their own. It’s still uncommon. Inmarsat Satphone Pro is a good phone, cheap with Internet and text capabilities. You may be able to use a shipmates satphone if you pay them.
Of course when you go ashore you can usually contact family and friends but underway is real hit and miss. A cellphone (not Verizon) set up for international calling and the old fashioned snail-mail will be your best bet.