I would like to first make the disclaimer that I learned most of these the hard way. I have been in numerous arguments with family members and friends who disagreed with my fundamental values and therefore fell victim to many of the heated screaming fights we all wish to avoid. After the results of the election came in however, it was disheartening to see that I could probably count on one hand how many people were having effective debates about policy. To help the flow of ideas travel throughout society I'm going to break down how to argue effectively and stay friends afterward.
Stop having value arguments
One of the things I learned in Speech and Debate was that there are two kinds of debates, value and policy. Value debates revolve around a persons fundamental belief system. An example of a value debate would consist of things like protecting human life is more important than protecting individual freedom or pizza the best food on the planet. These are dangerous debates to have and you need to stay away from them because there is never a "winner". You are never ever going to convince me that pizza is the best food on the planet so do not even try. You can talk to me about how the combination of sauce, cheese, crust, and customizable toppings make pizza untouchable by other dishes and I will scream at you that actually sushi is the best food on the planet and we will go around and around. Obviously this applies to more serious value debates. A Christian and an Atheist will never convince the other that they're right, but that does not mean they can't discuss policy or maintain a friendship.
Start having policy arguments
Policy debates are arguments over what approach to use to solve a problem. THESE are the types of arguments TO have. A Christian and an Atheist may not agree on why the human race exists, but they can come to an agreement or at least have an educated discussion on if prisons should or should not be privatized. These aren't arguments over the "best" type of prison system, the two would never agree. But by picking one type of prison system and discussing the pros and cons a healthy, effective debate about not only if they're effective, but what could done to decrease the amount of items on the cons list, is actually possible. Plus the two can likely stay friends afterward.
Keep in mind though, you cannot be a close minded, uneducated advocate of your side of the debate and expect to win and stay friends. If you haven't read up on the policy you're debating about, it's probably best to just listen because you'll have no choice except to resort to a value debate since your opinion is all you will be able to go off of. And if you are educated you are not an all knowing, all seeing, higher being that can solve all of the world's problems. You need to listen and engage even if you do not agree. Listening does not mean you agree with them, it means you have respect for the discussion.
Also remember to never yell (the first one who does loses), and to never interrupt. Keep in mind the more the other person says, the more fuel you have to cut down and oppose. I'll get into how keeping the other person talking is an effective strategy later. You have a duty as a productive and decent member of society to talk about the issues that are plaguing society with a polite and inclusive tone. If you alienate your "opponent" by calling them names, making them feel belittled, and turning against your cause then YOU JUST LOST. Remember why you're having these debates. It's not simply to spew off how knowledgeable and woke you are to the world so they bow down and do whatever you say. And it definitely is not to divide the country between us vs. them. If you sincerely want change to happen and a discourse to be open, you ALSO need to be open minded, educated, and a decent human being.
P.S. remember that this is much easier to accomplish when you have policy debates because value debates get really heated really fast because someone's fundamental identity is being attacked.
Tips for when you know you're right
This takes some maturity. Remember, with great power comes great responsibility. I for one will not ever try to debate anything sports related because I know virtually nothing about sports despite having been a cheerleader for 14 years. If I ever find myself in a discussion about sports I politely listen to it without offering my analysis because quite frankly I would sound stupid if I spoke up. Keep this in mind when you're debating. If you know virtually nothing about the crisis in Syria then when you find yourself in the midst of an argument you need to turn it into an educational, expository learning experience instead because without having first been educated on a topic you cannot then offer your opinion of it. And you will sound stupid to the other person because uneducated opinions are super easy to identify to a person who is educated on the issue.
With that being said, if you are educated on a topic and believe yourself fit to offer your opinion of it then that's awesome, but you're not entitled to have your opinions listened to and let alone adopted by anyone. Therefore it is MUCH more effective, at least in my experience, to have other people think your opinions are their opinions. The best way to do this is to ask questions as responses. Yes.. I'm telling you to use the Socratic Method to manipulate the person you're talking to into disproving their own argument, but there's a reason why this is so effective... it is non-threatening. "The Socratic method can be used to show someone that they are wrong, or at least imprecise, by getting them to agree with statements that contradict their original assertion". You will likely need to practice this to really get it down, but I promise once you do you will slay. It would take me a whole entire post to explain how to use this to you, and since there are plenty out there that have already done so I'm going to skip over it. A quick guide can be found here.
Also, do not be afraid to stop an argument that turns into one of value. For example, I am a hardcore environmentalist (not a purist though) and I believe that protecting the environment and ecosystems needs to be in the top 3 priories of every single person on the planet. So whenever someone tries to say "Global warming isn't real" or "there are more pressing issues than water pollution" I just politely say "I'm not interesting in discussing if these problems are problems, I believe that they are, I am interested in discussing their solutions so if you're not here to offer your input on policy solutions we can talk about the puppy bowl or Joe Biden memes instead". Nobody will ever convince me that pizza is better than sushi or that we do not have an obligation to protect the environment so why would I let them try and risk losing a friendship over it?
P.S. The deciding factor of if you stay friends or not after an argument is going to depend on your attitude during the argument.
If you want to change values you need to stay friends
Just because I value the environment and my family member doesn't does not mean I need to cut them out of my life. That's immature and irresponsible because then that's one less person in their life who does care. If you really want to change the way society behaves (I listed my 7 sins of society in this post) then you need to show them that they themselves are not society, they are a member in society. When your community and your circle cares a lot about an issue you begin to care a lot about it. If you cut yourself out of their world you have just cut out possibly the only advocate for your values in their community.
With that being said you need to make sure that if you're going to be the advocate for a cause you're doing so in an inclusive way. If you go on a hateful rant that separates you from those who disagree with you you've just now left no opening for them to come over to your side or to adopt your values. Change is really hard and you cannot change someone by trying to first hurt their ego, you need to be inclusive about it. To put it in perspective, it's really difficult to want to join a club when all of the members of the club are giving you dirty looks, calling you names, and think they're better than you. Keep this in mind.
Why I care
The reason why I care so much about how you're arguing is because I really, truly want solutions to start being discussed and dissected. I don't believe the solutions have been thought of yet and with the way I've been seeing people talk to each other after the election results, it doesn't look like any are in the near future. I do have faith though that if we shift the conversation to what we're going to do about the problems that are facing each and every community instead of who to blame for the problems then we will see real and positive change.