Evangelization is a word whose meaning comes from the Gospel. The persistent message of the Gospel is conversion and repentance. Last month the Pope said:
Evangelization is an immense mission, especially in this our time, in which humanity suffers from a certain lack of reflection and deep thought and in which a humanism is spreading that excludes God.... Because of this, it is still more urgent and necessary to illumine the new problems that arise with the light of the Gospel which does not change...preaching of the Gospel is an inestimable service that the Church can offer the whole of humanity that journeys in history.
The purpose of our lives is to reveal God to men. And only where God is seen does life truly begin.
We think of missionaries as the evangelizers. That is true; but we also are called to spread the Gospel wherever we might be. First, we should live as Christ teaches us, observing the Commandments, doing charitable works, etc. Showing the human virtues such as kindness and patience is most helpful in helping others to be open to the truth. Then, whenever the opportunity presents itself, we should speak of Christ, his Gospel message, prayer, the Sacraments, etc. This can be done in a gentle and informative way, never being argumentative or forceful. We can be the medium, but let God do the work.
The Holy Father also said that "...whoever participates in Christ's mission must inevitably face tribulations, rejection and sufferings, because he is confronted with the resistance and powers of this world..." and "...persecution is also proof of the authenticity of our apostolic mission."
So, opposition is often a sign that our efforts are good and that the truth is being heard even though the recipient does not want to hear it. It's kind of like saying to the speeder "Please, slow down," or to the drinker "Let me drive," and the response is "Leave me alone!" Saying the truth is done out of love and not out of self-righteousness, and we need to do it. As with everything, prayer is essential. With sincere prayer we will know what to say and how to say it.