Question: I feel very unsettled because I want to feel free and happy but have never been able to be not affected by other people's suffering. Please tell me what I can do.
Response: The very nature of existence in samsara, the cycle of rebirth, is suffering. It's the first of the four noble truths--life entails suffering. Every being in samsara undergoes suffering to some degree. Even in the highest heaven, the beings undergo suffering when their lives there are about to come to an end.
So first, we need to accept the existence of suffering and not feel that we are when others are not. Once we understand that it's a part of life, we can stop railing against it and move on to the next three of the noble truths: suffering is caused; it has an end; and the way to its end is self-discipline, concentration, and wisdom.
To no longer be affected by other's suffering, we practice non-attachment. For more on this, please read Non-attachment Not Detachment.
Also, to truly help others, we need wisdom, not just compassion. The last entry Compassion Tempered with Wisdom might be helpful on this.