Ask HN: What's currently the best way to get global phone service?

I'm looking to upgrade my dinosaur-age mobile phone to an Android. I'm currently living in the States, but I'll be in Puerto Rico for a couple of months during the winter and going to Hungary for a year or so in the spring. I don't know when I'll be back in the States. Never, maybe. Or maybe next fall, just depending on how life turns out.

The question: is it better just to get new service in Hungary, or is it practical to have a global service of some kind?