There are different types and motivations involved in arguing. Giving attention to the latter is prone to be more effective at achieveing a healthy practice of it.
Here are several motivations:
You want to win.
You want to stand up your ground (usually these are counterpoints.)
You want to get to the truth.
The last one should always be the end goal. It may not always be achievable, but at least you have your best shot with desire for truth as a starting point.
We can argue about lots of things, objective and subjective:
Is there a God?
Is blue the best color to paint the house?
Do you love me?
Was Dick Butkus was a better linebacker than Lawrence Taylor?
Is less government better?
Does light travel in waves or particles?
Truth of course, as Pilate brought to our attention is often a dicey thing, but I am not talking here about any specific truths, or in fact how to define truth, but demanding that if we have any shot at getting there through argument, then getting there must be our motivation.
Let me use a legal trial as an example. Are the opposing lawyers trying to establish whether in fact the accused is guilty or innocent? Or are they trying to win? Most of us would say the latter. But is that the best way of getting to the truth? No.
Often because of our insecurities or stubbornness, we simply try to win our case. That will always stifle a dialogue which is a necessary tool for growth.
Listen to the other side with an open mind.