Seek the Middle
Discussing hot-button issues is extremely challenging, and one reason for this is that everyone enters such conversations expecting to fight. The fact that these conversations happen isn’t a terrible thing — the alternative would be for everyone to think what they think and make no progress.
I think we often fight for the wrong thing, however. Instead of fighting to prove ourselves right, we should be fighting for the truth. It’s rare that big national/political issues aren’t based at least partially in fact on both sides. Seek out these facts, see how both sides are using or abusing them, and form an opinion that lives outside of team A or team B.
This isn’t common, and will lead to frustration. It seems generally true to me that if you say something leaning one way, the person you bring it up to will assume you lean ALL the way in that direction. Being good at debate/conversation/whatever requires looking at the middle.