I wish more people would focus more on the context behind people's opinions. By taking time to understand the experiences that shape people's values and priorities, I've been able to better anticipate how they'll react in new situations. So, when given the opportunity, I try exchanging childhood stories with people that I'm having a hard time understanding. Frequently, after hearing some of their formative stories, I get why they respond the way they do, even if I disagree.
While this sounds obvious, people are quick to pass judgment on others, often without even trying to understand their motivations. When this empathy is extended in both directions, it's much easier to reach compromise. By taking time to understand why a reasonable person with a different set of life experiences could disagree with me, I'm better equipped to start a productive dialog.