How do I change one of the cities that I have on my Weather app?

Open the Weather app on the iPhone, tap the list icon at the bottom right corner of the screen, slide the city you want to remove to the left and tap the trash icon. To add another city, use the Search bar near the top of that same screen.

The above instructions are for the current iOS version but if I remember correctly, will also work with iOS 10.

See this from the iPhone User Guide.

Check the weather in other locations on iPhone - Apple Support

To delete New York, from this screen, swipe New York from right to left and tap the resulting trash bin.