Travelling is an opportunity to gain insight into the multitude of cultures that exist around the world; when experiencing the customs of the country that you're visiting, it becomes so much more than just a holiday.

Immersing yourself in local traditions is an excellent chance to learn more, not only about others, but also about yourself. You could discover a new favourite artist while at a foreign music festival, find a new amazing dish whilst eating at a local restaurant, meet people you never would have met if you hadn't practiced speaking the native language at the bar, or, as in my case, uncover a fascination for flamenco after attending a traditional show!
...speaking English accompanied by hand gestures just isn't going to cut it.
As a languages student, I believe one of the best ways to honour the local culture is to make an effort to learn a little bit of the native language - a simple "hello" or "thank-you" can go a long way. Visit any smaller European town that isn't Paris or Barcelona and you'll quickly find that speaking English accompanied by hand gestures just isn't going to cut it. Asking for the bill becomes a complicated ordeal that could have been avoided if you had taken time to look up the word before arriving at the restaurant.

Speaking of cuisine, instead of allowing yourself to be pulled toward the huge restaurant with an English menu, spend a bit of time exploring the backstreets; before long you'll come across somewhere smaller and perhaps family-owned.
Not only will this save you money, it comes with an authenticity that you just won't find in those tourist-trap eateries in the main square - and your money will go towards supporting the local economy.

With just a few changes, you are able to truly experience another culture...
Being a respectful tourist is simpler than it seems; all it takes is learning a few greetings in the native language, engaging with the local people, and trying things you perhaps wouldn't have thought to before. With just a few changes, you are able to truly experience another culture - and there isn't anything more rewarding than that while travelling!