How to talk with folks from the other view

My teenager chided me for bringing up the topic of the inauguration with one of her companions who is from a conservative home. Of course I knew her family's orientation. I wanted to open dialog. The companion didn't want to engage, so it didn't go anywhere (except my kid chiding me).

How should I open dialog with those whose views differ from mine? What I want is for us to look forward and find common ground. Positive ground. How? Maybe I should pose a question like, "What are your hopes for the next year?" or "next administration?"

Or perhaps I could share one of my favorite lines from the inauguration speech (or the invocation!!) and ask if others find inspiration in it as well.

I hope I can make some headway. I really want to make a little more peace in the real world.

Peace, y'all

Molly