How do I enter Thailand with a new passport and a valid visa in my old passport?

Doesn't matter what your nationality is as it works the same for everyone..

IF you get a new passport while you're out of the country and you have an extension of stay and a re-entry permit in the old passport. You show BOTH passports when you show up at passport control to stamp IN to thailand and they will stamp you in on the new passport until the expiration of your extension.

THEN

You go to the immigration office where you got the extension issued with both passports and they will move the extension and any valid re-entry permit to the new passport for you.