Forgiveness is tricky. It’s very difficult to get from a place of anger and resentment to forgiveness. The best way I know how to forgive is to try to find empathy for the respective person or people–attempting to put myself in their shoes and see the world from their point-of-view. This is, of course, a process and it takes time, patience and fortitude (and the amount of time is commensurate with the level of resentment), but it’s usually worked for me.
With children, apologies are so simple. Usually a grown up–a parent or teacher, for example–say when it is called for: “say you’re sorry for hurting Johnny’s feelings.” Oh, OK. And then it happens, perhaps begrudgingly or with a pout, and 1-2-6 it’s over. Everyone is back at the sand table playing next to one another until the next infraction.