Вам нужно свойство кнопки relief - SUNKEN и RAISED
Говнокод:
def change_but1():
but2['bg'] = '#ffffff'
but2['relief'] = tkinter.RAISED
but1['bg'] = '#eeeeee'
but1['relief'] = tkinter.SUNKEN
def change_but2():
but1['bg'] = '#ffffff'
but1['relief'] = tkinter.RAISED
but2['bg'] = '#eeeeee'
but2['relief'] = tkinter.SUNKEN
but1 = tkinter.Button(root, text = "Привет!", command=change_but1)
but1.config(bd = 1, font = ("Castellar", 25), bg = "white")
but1.place(x = 50, y = 160, height = 50, width = 150)
but2 = tkinter.Button(root, text = "Пока!", command=change_but2)
but2.config(bd = 1, font = ("Castellar", 25), bg = "white")
but2.place(x = 50, y = 220, height = 50, width = 150)