We're currently testing out Project Fi from Google, which is only available to residents of the United States at this time. And once you leave the States, it offers the same cheap rates no matter where in the world you're using it from. We're trying it out as we run around Europe this summer, and we'll report back when we have further advice to share.
Previously, our recommendation was the Moto G, which is now in its fourth generation. This model has been considered by reviewers (see our archived article) an excellent price-conscious choice. It runs Android 6.0 and comes unlocked, so you can swap in a local SIM card in each country you visit so that you have a local phone number and 4G internet access anywhere.