I try to learn the following sentences in the language of a country I go to if I don’t know anything at all about that language:

• Hello, as polite as possible
• Goodbye, as polite as possible
• A beer please
• I’m sorry I don’t speak <language>: do you speak English?

I find this quite enough to strike up a conversation in most of the world. When people don’t automatically switch to English - perhaps because they don’t know it well enough - then I try another “universal” language like Spanish or French (universal mostly because of past colonialism, sadly). That implies speaking those languages of course.

If the locals won’t speak in English because of a prejudice against English (mostly French-speaking regions) I don’t even bother with the “I’m sorry…” bit.

If the locals are anti-Americans - very common, and getting more and more common - I affect a British accent. I wouldn’t fool a Brit for a New York minute but I’m convincing enough for a non-native English speaker to avoid being associated with the US.