I used to try to learn a few phrases in the local language but everyone in tourism or even retail speak a little English now.
I am reasonably fluent in Spanish after living in Arizona most of my life and taking 2 years of Spanish in college. I found that in Spain a couple of years ago that I could get by fairly well and I have no doubt if I stayed there for a month or two I would be fluent. And Spanish gives me a pretty good leg up on at least reading French, Italian, Portuguese, Catalan, and even Romanian.
I speak Swedish almost fluently which lets me mostly understand written Norwegian, Danish, and to some extent Icelandic. Spoken Danish and Icelandic are tougher but if I speak Swedish to them then understand and usually modulate their own spoken language to mimic Swedish and it works.
Plus I took German in high school and, while it is not that close to Swedish, especially the grammar, I can mostly read German and get my point across because there are a huge number of cognates. And Swedish+German=Dutch (almost). So I do pretty good in western Europe.
But I always try to learn a few words when I go somewhere - please, thank you, I'm sorry, excuse me, I need help, etc. I try very hard not to be an ugly American and expect others to deal with me in English. Google translate has become an enormous help recently!
I have had some interesting experiences trying to get things done in various countries including finding tools and special hardware in Japan when working on a project there. It never ceases to amaze me how well humans can communicate without language by miming, sketching, and pointing.