When Savannah and I first moved in together, there was a learning curve. We got on each other's nerves a lot. Then, things settled down. We learned a lot about ourselves during that time and I know we are both better people because of it. I think that is the biggest thing I can advise about conflict with people you are in close relationships with- learn from it.
Most arguments are rooted in pride from both parties. Even if someone has wronged you and you are the victim, there is still a little pride on your part because you don't feel like you deserved to be treated that way. I'm not saying you did deserve to be treated any kind of way. But real talk- we all deserve hell, so there's that. We have no right to be treated a certain way. We have no rights. We are slaves to The King. I think when you step away from the argument and accept that in some way you contributed to it (regardless of the origins of said argument), both parties will benefit.
When I am upset with someone, my mom tells me that I need to ask God what this reveals about MY heart. Ugh. I hate when she says it because when I'm mad at someone, I just want to be mad at them. I don't want to self-reflect. I just wanna wallow in how this person wronged me. However, her perspective has drastically changed how I handle conflict with people. Why am I so upset about what this person did to me? There is always a lesson to be learned there.
If I'm honest, I can't really tell anyone how to argue better. There is definitely a list of do's and don'ts to follow, but when you're mad- all of that seems to go out the window. It's hard to be reasonable when you're angry. Obviously avoiding being angry is a great first step, but sometimes that's just not possible. So you might not be able to control getting angry in the first place or what you do or say when you're angry (within reason of course, violence is obviously controllable), but you can absolutely control how you handle the aftermath. Relationships can sustain any number of arguments as long as there is proper reconciliation afterwards.
Also, if you can, pray a quick little prayer in the midst of your argument. If that fails, definitely pray about it once you've stepped away and cleared your head. God will let you know what to do, but I promise you He will ask you to apologize for your part. You have to talk the argument over with the other party involved. One of the worst things you can do is move on without talking about it. That leads to a much bigger argument down the road. Deal with it now, so you can move on.
Fighting sucks, but it's a part of life. How you deal with fighting will be a major factor in the health of your relationships. If you truly care about the person you are fighting with, then you have to look at them through God's eyes and remember that no one is perfect. Whatever this person has done, you've done something just as bad in the eyes of God. Keeping God in the midst of conflict is how you will heal and move on and become a better person.