Hey Heather and Lars!
I’m Polish, who spent summers in Europe. Similarly, I mostly use my phone for internet connection and the mobile operator market is changing so fast that every time I come back to Europe I choose the current best offer.
Recently, the EU forced all mobile operators to allow “free roaming”, which on paper means that whatever you have in your base country, you’ll get everywhere else in Europe. In practice, all operators retain to Fair Usage Policy, which in short means that after 4 consecutive months, they’ll look back and check if you used your plan more in your base country or abroad. If abroad, they’ll charge you extra fees.
Most EU markets are controlled by big telcos like T-Mobile, Vodafone or Orange, but they focus on selling long-term contracts with some “European plan”, but I’d avoid it. Their offer is usually very complex, expensive and you’ll still be restricted by Fair Usage Policy of the country, where you signed-up the contract.
The only viable option, in my opinion, is to buy a pre-paid plan, that’s usually for 30 days and in case it’s not what you expect, buy another one.
I’ll be travelling between Poland and Italy a lot this summer and the most convenient offer I found so far will be Lycamobile. A virtual operator focused on international customers. They offer quite big data quota in addition to the EU phone calls, that I’ll need for my work.
Lycamobile is present also in the USA. Check their plans and compare them to what you’ll get with their European card with a new number. You can also buy a new Lycamobile every 3 months in your new “base country”.
As for calling to the US, I’d use Skype, VoIP or Hushed app on your phone. That’s in most cases cheaper and faster than calling via regular telephone infrastructure. There’s also a way to “connect” your current American mobile number to Skype in order to receive calls, but you’d need to double check it with your current provider.
Hope that helps
PS. Please check details on how Schengen zone and travelling visas affect your allowed stay. All the countries you mentioned are already tied to the Schengen rules.