This trip will be a little more complex (financially speaking) since there are multiple currencies involved, and your child will be managing a new currency in each country (though in my experience, many places in these countries will accept Euros, though the rate will be set by the merchant).

I would still suggest taking an ATM card, credit card, and approx $50-$100US cash for emergencies. The nice thing about traveling in a group is that should something dire happen (lost ATM card, theft, etc), your child will have people he/she can lean on for a few days until a card is replaced.

I would also talk to your child and develop a strategy - budgeting how much to take out of the ATM in each country based on how long the group will be there, with the goal of minimizing the amount of local currency on hand when they move from one country to another.