My youngest daughter just turned 6 yrs old. We've found that being consistent, consistent, consistent with rules and consequences works best. We have some rules that cover all the bases of unacceptable behavior.
1. Be kind.
2. Be respectful.
3. If you're not sure about something, ask first.
4. If you're not at home, follow the rules of the place you're a guest in.
5. Sharing is always a good thing.
6. Be courteous.
7. Listen to others speak and take turns in conversations.
If you notice, I try not to create "no" rules. I try to create positive rules. The reason for that is that my eldest daughter is ODD (Oppositional Defiant Disorder) and my middle daughter has Autism and if you present "no screaming, no running, no hitting/biting/kicking" as rules then you've pretty much given them ideas for behavior for them to engage in that will push buttons. If you present positive rules there's a better starting point for what's acceptable and more chances to praise for following the rules.
When a rule is broken and someone has been unkind or disrespectful, I get down face to face with them but not intimidating and tell them in Mommy voice that Specific Behavior is unacceptable because it's not (whatever rule they broke) and that they're getting a warning about which rule they've broken and a chance to correct that behavior. They're also warned that if they repeat the behavior/break the rule again (within a short period of time) then they're going to have a consequence (like a time out). If they choose not to correct the behavior then they're removed from the situation and they get a "time out" or "cool off thinking time." After the cool off time is over, which is always the age of the child in minutes, we have a quick, "Do you know why you were in time out?" "kid gives explanation" "Okay, I'd like an apology." "kid apologizes" then hugs all around. Lather, rinse, repeat until they get it.
If you don't use time outs, such as with my older daughter as time outs aren't always a good enough incentive, we've found that losing a privilege is good incentive. We've also found that if she loses a privilege that earning it back is also good incentive. It's far more effective than groundings and she always has something to work toward.
In any situation, keep the conversation short and simple. No arguing. don't drag it out. Kids think that the more you argue the more chance they have of "winning." I always listen to their side, let them know I understand and when they have a valid point, but once my mind is made up and a consequence is given then the discussion is over and I stop engaging.