I've found that Mexicans (especially outside of major tourist areas, but even there) LOVE it if visitors attempt to speak Spanish with them, even just a few words.

Thailand is pretty similar. I've lived here for ~10 years and can speak Thai fairly well. So, many locals know me and aren't surprised when I speak Thai with them, but if I travel I get a lot of smiles just like this video.

I guess French people are stereotypically less patient/pleased to deal with visitors trying to use the local language, but I don't know if that's true. Never been there, unless Quebec counts (where it didn't seem true).

No matter where I go I always learn a few short phrases out of respect, and it pretty much always brings a smile.

The first and most important is 'thank you' and, while 'yes', 'no', 'excuse me', and 'hello' are good, the second phrase I always learn is how to ask for the bill (or the 'check', as I learned before going to the states). Pretty much anywhere you go you'll probably eat out, there's no response needed (beyond 'thank you'), and it doesn't hurt that it sometimes means the bill comes with a complementary drink.

I've had a pretty good response to my butchering of french in Montreal, The shop keeper caught on right away and let me struggle a bit then saved me by speaking english. I think rural areas are less forgiving.

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Speaking even a little Italian has brought smiles, extra plates, great service, and complete confusion (I'm not caucasian); same with Chinese.
Gotta work on my Spanish!

I recently visited Barcelona. My Castellano is decent but I know almost no Catalan, and was getting along fine, but one evening I went to a restaurant that served authentic Catalan cuisine. (The street was plastered in Catalan flags!) I spent a bit of time that day learning a few Catalan phrases ("Good evening", "I have a reservation") etc, and my efforts were definitely appreciated, as we very warmly received by the staff, who were courteous enough to switch to English (with a little good-natured ribbing) when my Catalan well ran dry. Big smiles (and great food) all round. It definitely makes a difference.

In Sweden I got in many language battles. In the shops, post office, on the street, with friends... I would speak Swedish, they would hear my accent and reply in English, and it would continue like that.

I was asked many times why I persisted in speaking Swedish when we could converse in English.

Because I came to YOUR country. I owe you that respect.