Should I visit my dad without my wife?
That is strange. Why do you want to visit without your wife?
There are many reasons, my wife is always nagging me all the time.
You don't have to spend all the time with you wife but it is expected to be with her when visiting family. Your family is now her family and not visiting with her could be considered rude by others.
Nah, it will be okay. I'm going to go by myself.
It is not good to do something that will upset your wife. She should be invited to spend time with your family and with you so that they can become closer.