1. Try to stay in one place for a few months, minimum. If you can stay a year or two, do that. Jumping from city to city every week seems exciting at first, but it quickly makes everywhere feel the same.
2. To quote Lao Tzu, when your work is done, forget it. If you work remotely, actively disconnect at the end of the day. It's too easy to be constantly plugged into the English-speaking media world and ignore what's right in front of you.
3. Try to blend in and adopt local clothing, culture, foods, etc. Read books by local authors, watch local films. A bit obvious, maybe, but I've met many people who insisted on only eating Western food, reading Harry Potter, and watching Netflix while abroad. At that point, why even bother traveling?