In the following example there are 2 Screens: SettingsScreen and MenuScreen
Using the first button, on the current screen will change your screen to the other screen.
Here is the code:
from kivy.app import App
from kivy.lang import Builder
from kivy.uix.screenmanager import ScreenManager, Screen
# Create both screens. Please note the root.manager.current: this is how
# you can control the ScreenManager from kv. Each screen has by default a
# property manager that gives you the instance of the ScreenManager used.
Builder.load_string("""
<MenuScreen>:
    BoxLayout:
        Button:
            text: 'First Button on Menu'
            on_press: root.manager.current = 'settings'
        Button:
            text: 'Second Button on Menu'
<SettingsScreen>:
    BoxLayout:
        Button:
            text: 'First Button on Settings'
            on_press: root.manager.current = 'menu'
        Button:
            text: 'Second Button on Settings'
""")
# Declare both screens
class MenuScreen(Screen):
    pass
class SettingsScreen(Screen):
    pass
# Create the screen manager
sm = ScreenManager()
sm.add_widget(MenuScreen(name='menu'))
sm.add_widget(SettingsScreen(name='settings'))
class TestApp(App):
    def build(self):
        return sm
if __name__ == '__main__':
    TestApp().run()