All of these ports will take US $ - when you are on a cruise ship this is a lot easier than trying to get the local currency. You will probably get change from local merchants etc in the local currency. Belize will NOT have the same currency as Mexico although due to their proximity to Mexico, you MAY find that some merchants will take Mexican pesos. My advice for a cruiser - just take US $. If you were staying just in Mexico, my advice would be different. And please do not be obnoxious when you are given your change in pesos (not that I think you will be but I have seen tourists in Mexico berating local merchants for not giving them change in US currency). Have a great trip !
The Belizian Dollar is valued much higher than the Peso. You do not need to change your currency. Make larger ($20) purchases with credit card. The Pharmacy which sells prescription medications over the counter takes cash. They have price lists on the wall. There is a cigar store at Belize too.
Bring small bills - $1's, $5, $10 as taxi drivers don't like to make change. This will save you money in the long run.