I'd also add, "eat everything in sight". I've found that more than anything, nothing can symbolize that you respect a culture more than eating their food with them, respectfully enjoying it and just trying to have a good time. It shows an interest beyond superficials and most people enjoy sharing something about their culture with you through food.
You are what you eat, and if you eat what they eat, you are in some sense made of the same stuff.
Humans like to put other humans in two boxes, "us" or "them" and nothing can get you over the threshold and into the "us" box better than eating their food. In my experience, I've also found that most people want people in the "them" box to be in their "us" box and will lightly test your suitability with various social normalization tests -- often in the form of some local delicacy that they know foreigners won't be into. "Here! Try some of this delicious fermented cow stomach!"
(it also helps that most places have a friendly social structure around alcohol consumption, and chasing bites of random animal innards with some hard booze seems to do a good job of killing off whatever might not agree with you)
It also usually follows that they'll see you put a good effort into meeting them (culturally) and most folks will give you a wide berth to stumble through their social customs.
Without knowing anything else about a place, I've managed to muddle through relations with Peruvians, Ecuadorians, French, Russians, Chinese, Koreans, Saudis, Germans and a few others simply by sitting down at a table with them and breaking bread and trying to mind my manners.
In a few places, this immediately broke down walls of outward hostility and the night often ended with some kind of embarrassing duet at a Karaoke bar or similar.
That does help. My dad was in the army and was stationed in Korea for a few years. One weekend he was eating lunch with some Koreans he would be working with. Well, near the end of the meal he was offered some burnt rice. He graciously accepted it and ate it. After the meal his translator informed him that burnt rice was considered a delicacy in Korea. Furthermore his predecessor had been offered the same dish and refused. His predecessor lost a lot of respect by that decision