Onboard ship - you want to give them a credit card at the start of the cruise to cover all of your onboard expenses. Do NOT use a debit card because they will usually put about a $500 "hold" on funds in your acct on the first day, and that can affect how much money you have to spend on the cruise.
In the Caribbean you can use dollars almost everywhere (or credit cards). Sometimes vendors prefer cash dollars over credit cards if there is a large fee, but usually it makes no difference. You get more protection making purchases with credit cards.
In Europe - you can always use your ATM card to get local currency in any country from an ATM. Yes, they take almost any US bank's debit card (they are all on the same "Star" or "Cirrus" banking networks) The surcharge is not too bad (better than a currency exchange storefront).
In mexico - most vendors will take dollars or credit cards.
In Canada - generally they prefer that you go to the bank and get Canadian Dollars, or you can use credit cards.
Alaska & Hawaii: really?