How do I permanently change my location In the weather app?

Perhaps your refer to a Weather Widget. If so, touch-and-hold (i.e., long-touch) the Widget…

To change the displayed location to another city, select Edit Widget, tap Location then select the preferred city or Current Location.

If instead you are attempting to change the list of cities of interest within the Weather App itself - open the Weather App; within the associated sidebar you will see a list a locations for which a forecast has been configured. At top-right of the Weather sidebar, tap the ellipsis (“…”) icon - and from the menu tap Edit List. 

Each of your configured locations will now have an associated red “-“ button. Tap the button for each location that you wish to delete from the Weather App. When you’re finished, tap Done.

You’re very welcome.