I think the key is to be friendly and not judge people and not come across as a fanatic. And be patient - don't expect any immediate results, lifestyle changes don't happen overnight. It may be important to leave emotions out of this, though I am not sure about that.
Reflecting on my own experience, I don't remember anyone telling me explicitly what I should do with my lifestyle, but I did meet several people who were talking about climate change and their own personal choices. The discussions were always calm and intelligent. In hindsight I can tell those conversations were a contributing factor in triggering my own lifestyle changes. But it took years.
Bringing up the latest climate news piece could be a good entry point for a conversation. Before I joined my current project team I'm quite sure they never talked about climate. Now we regularly discuss it during lunch breaks.
