We generally just pick up a SIM in the new country. Cheaper and simpler than trying to get a worldwide cell plan.
The only exception would be if you are bouncing around to a new country every few days (EU for this purpose would generally be considered one country as I believe they mandated coverage across all member
Other than the USA, which is a much bigger hassle, in most countries at the airport
or any mobile shop, for $10-30, they will pop a new sim card with 10-50GB for the month in your phone
and get you up and running within 10-15min.
If you need a permanent number, get an IP number (like Skype but many alternatives if you don't like them)