Listen without trying to cure. Being truly present to a grieving person, without judging or denying their pain, without trying to fix the unfixable, is the single most important gift you may give someone who is grieving.
Understand that this will take time. This is not like a cold or a cut that will continuously get better. Grief will come in unexpected waves, affecting a person throughout their lifetime. Each person, even within a family, will have a different time schedule and, sometimes, it gets worse before the intensity and frequency of the waves change.
Be direct and honest. Use real language (e.g., "he died") versus euphemisms such as "passed away" … "gone to a better place" … "not with us any more" …etc. In our well-intentioned way, we are frequently overprotective and tip-toe around issues of death and dying. This language especially confuses children. "If he's in a better place, why am I not there?"
If you don't believe it, or you don't have an answer, don't say anything. It is always better to remain quiet, or admit what we don't know, rather than responding to fill the space. This will create a sense of trust between you and the person grieving.
Don't avoid topics that include the deceased. After a death, we need outlets to create the significance of the person who died. We do that through talking about that person, memories and legacies.
Allow for humor and lightness. Grieving people need breaks from their grief, and time to remember what it feels like to be 'normal' while adjusting to the 'new normal.'
Take the lead from the person who is grieving. In public places (supermarkets, school grounds, etc.), reflect their mood and their words, allowing them the room to be where they are at the moment.
Be aware of your own limitations. Many people cannot sit with another person in pain, especially when there is nothing functional that can be done. There are many ways to support the grieving person - assisting with childcare, distracting them with movies and meals, offering a second home, helping with the household bills and mail, etc.