I made the following changes to your code to get it working:

Gridded the root window so that container would occupy the entire space in the window.

#Gridding the tkinter window
self.grid_rowconfigure(0, weight = 1)
self.grid_columnconfigure(0, weight = 1)

Made container expand in all directions using sticky = 'nsew'.

container.grid(row=0, column=0, sticky = 'nsew')

Modified __init__ and added tkraise to all the Page classes.

I will use the below block of code for explanation purposes.

class StartPage(tk.Frame):
    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        
        # label of frame Layout 2
        welcome = ttk.Label(self, text ="Welcome!", font = LARGEFONT)
        welcome.grid(row = 0, column = 2)
        
        #Creating a frame exclusively for the buttons
        self.frame_buttons = tk.Frame(parent)
        self.frame_buttons.grid(row = 1, column = 0, columnspan = 3)
        self.frame_buttons.grid_remove()
        
        #Gridding self.frame_buttons
        self.frame_buttons.grid_columnconfigure((0,1), weight = 1)
        self.frame_buttons.grid_rowconfigure(0, weight = 1)
        
        button1 = ttk.Button(self.frame_buttons, text ="Page 1", command = lambda : controller.show_frame(Page1))
        button1.grid(row = 0, column = 0,  padx = 10, pady = 10)

        button2 = ttk.Button(self.frame_buttons, text ="Page 2", command = lambda : controller.show_frame(Page2))
        button2.grid(row = 0, column = 1, padx = 10, pady = 10)
        
    def tkraise(self):
        self.frame_buttons.grid()
        self.frame_buttons.tkraise()
        tk.Frame.tkraise(self)

In the __init__ method, self.frame_buttons is created and all the buttons are added to it. It is displayed using .grid and then immediately removed with .grid_remove(). The advantage of using .grid_remove() is that the grid options are remembered and we can directly use .grid() to display the frame in future.

In the tkraise method, the frame is displayed, lifted to the top and then tk.Frame.tkraise(self) is used to call the in-built method.

Working Code:

from cgitb import text
import tkinter as tk
from tkinter import ttk
  
LARGEFONT =("Verdana", 35)
  
class tkinterApp(tk.Tk):
     
    # __init__ function for class tkinterApp
    def __init__(self, *args, **kwargs):
         
        # __init__ function for class Tk
        tk.Tk.__init__(self, *args, **kwargs)
        
        #Gridding the tkinter window
        self.grid_rowconfigure(0, weight = 1)
        self.grid_columnconfigure(0, weight = 1)
         
        # creating a container
        container = tk.Frame(self) 
        container.grid(row=0, column=0, sticky = 'nsew')
  
        container.grid_rowconfigure(0, weight = 1)
        container.grid_columnconfigure(0, weight = 1)
  
        # initializing frames to an empty array
        self.frames = {} 
  
        # iterating through a tuple consisting
        # of the different page layouts
        for F in (StartPage, Page1, Page2):
  
            frame = F(container, self)
  
            # initializing frame of that object from
            # startpage, page1, page2 respectively with
            # for loop
            self.frames[F] = frame
  
            frame.grid(row = 0, column = 0, sticky ="nsew")
  
        self.show_frame(StartPage)
  
    # to display the current frame passed as
    # parameter
    def show_frame(self, cont):
        frame = self.frames[cont]
        frame.tkraise()
  
class StartPage(tk.Frame):
    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        
        # label of frame Layout 2
        welcome = ttk.Label(self, text ="Welcome!", font = LARGEFONT)
        welcome.grid(row = 0, column = 2)
        
        #Creating a frame exclusively for the buttons
        self.frame_buttons = tk.Frame(parent)
        self.frame_buttons.grid(row = 1, column = 0, columnspan = 3)
        self.frame_buttons.grid_remove()
        
        #Gridding self.frame_buttons
        self.frame_buttons.grid_columnconfigure((0,1), weight = 1)
        self.frame_buttons.grid_rowconfigure(0, weight = 1)
        
        button1 = ttk.Button(self.frame_buttons, text ="Page 1", command = lambda : controller.show_frame(Page1))
        button1.grid(row = 0, column = 0,  padx = 10, pady = 10)

        button2 = ttk.Button(self.frame_buttons, text ="Page 2", command = lambda : controller.show_frame(Page2))
        button2.grid(row = 0, column = 1, padx = 10, pady = 10)
        
    def tkraise(self):
        self.frame_buttons.grid()
        self.frame_buttons.tkraise()
        tk.Frame.tkraise(self)

# second window frame page1
class Page1(tk.Frame):
     
    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        label = ttk.Label(self, text ="Page 1", font = LARGEFONT)
        label.grid(row = 0, column = 2)
        
        #Creating a frame exclusively for the buttons
        self.frame_buttons = tk.Frame(parent)
        self.frame_buttons.grid(row = 1, column = 0, columnspan = 3)
        self.frame_buttons.grid_remove()
        
        #Gridding self.frame_buttons
        self.frame_buttons.grid_columnconfigure((0,1), weight = 1)
        self.frame_buttons.grid_rowconfigure(0, weight = 1)
        
        button1 = ttk.Button(self.frame_buttons, text ="StartPage", command = lambda : controller.show_frame(StartPage))
        button1.grid(row = 0, column = 0,  padx = 10, pady = 10)

        button2 = ttk.Button(self.frame_buttons, text ="Page 2", command = lambda : controller.show_frame(Page2))
        button2.grid(row = 0, column = 1, padx = 10, pady = 10)
        
    def tkraise(self):
        self.frame_buttons.grid()
        self.frame_buttons.tkraise()
        tk.Frame.tkraise(self)
  
# third window frame page2
class Page2(tk.Frame):
    def __init__(self, parent, controller):
        tk.Frame.__init__(self, parent)
        label = ttk.Label(self, text ="Page 2", font = LARGEFONT)
        label.grid(row = 0, column = 2)
        
        #Creating a frame exclusively for the buttons
        self.frame_buttons = tk.Frame(parent)
        self.frame_buttons.grid(row = 1, column = 0, columnspan = 3)
        self.frame_buttons.grid_remove()
        
        #Gridding self.frame_buttons
        self.frame_buttons.grid_columnconfigure((0,1), weight = 1)
        self.frame_buttons.grid_rowconfigure(0, weight = 1)
        
        button1 = ttk.Button(self.frame_buttons, text ="Page 1", command = lambda : controller.show_frame(Page1))
        button1.grid(row = 0, column = 0,  padx = 10, pady = 10)

        button2 = ttk.Button(self.frame_buttons, text ="Startpage", command = lambda : controller.show_frame(StartPage))
        button2.grid(row = 0, column = 1, padx = 10, pady = 10)
        
    def tkraise(self):
        self.frame_buttons.grid()
        self.frame_buttons.tkraise()
        tk.Frame.tkraise(self)
        
# Driver Code
app = tkinterApp()
app.geometry("800x425")
app.mainloop()

Causing a widget to appear requires that you position it using with what Tkinter calls "geometry managers". The three managers are grid, pack and place. Each has strengths and weaknesses. These three managers are implemented as methods on all widgets.

grid, as its name implies, is perfect for laying widgets in a grid. You can specify rows and columns, row and column spans, padding, etc.

Example:

b = Button(...)
b.grid(row=2, column=3, columnspan=2)

pack uses a box metaphor, letting you "pack" widgets along one of the sides of a container. pack is extremely good at all-vertical or all-horizontal layouts. Toolbars, for example, where widgets are aligned in a horizontal line, are a good place to use pack.

Example:

b = Button(...)
b.pack(side="top", fill='both', expand=True, padx=4, pady=4)`

place is the least used geometry manager. With place you specify the exact x/y location and exact width/height for a widget. It has some nice features such as being able to use either absolute or relative coordinates (for example: you can place a widget at 10,10, or at 50% of the widgets width or height).

Unlike grid and pack, using place does not cause the parent widget to expand or collapse to fit all of the widgets that have been placed inside.

Example:

b = Button(...)
b.place(relx=.5, rely=.5, anchor="c")

With those three geometry managers you can do just about any type of layout you can imagine.

Actually in the previous semester I have also made some Tkinter application which is the project given by teacher to us. So I go to some tutorials website of Python and find three methods of placing the widgets on the output screen. The three methods are

 1. Pack()
 2. Grid()
 3. Place() #the best method over Pack() and Grid()

Place() method take the coordinates in the form of the x and y. See this link for more clarification https://www.tutorialspoint.com/python/python_gui_programming.htm

https://www.tutorialspoint.com/python/tk_place.htm

See the bottom of the Page of the given link.The Place() method is defined with its proper arguments. I will prefer the Place() method over Pack() and Grid() because it works like CSS as we use in html, because it take the value of (width,height) and place your widget according to wherever you want.

If you find your answer a thumbs up will be appreciated.

You can pass the row and column by using a lambda expression for the button:

button = Button(..., command=lambda row=i, column=j: doSomething(row, column))

To center horizontally this should be enough

button.pack(side=TOP)

But to center horizontally and vertically you could use nested frames. Check the following script:

import tkinter as tk

#%% Frames
frameA = tk.Frame(background="#c8c8c8")
frameB = tk.Frame(width=200, height = 200, background="#646464")
# Nested Frame. framebb is created within frameB without width or height
framebb = tk.Frame(frameB, background="#646464")
frameC = tk.Frame(width=100, height = 100, background="bisque")

frameA.pack(side='top', fill=None)
frameB.pack(side='top')
# expand is the key parameter to center the framebb within frameB
framebb.pack(expand=True)
frameC.pack(side='bottom')

#frameA.pack_propagate(False)
frameB.pack_propagate(False)
frameC.pack_propagate(False)

#%% Buttons and Labels
tk.Label(frameA, text = "Text within the frame A").pack()

a = tk.Button(framebb, text = "A").pack()
b = tk.Button(framebb, text = "B").pack()
c = tk.Button(framebb, text = "C").pack()
d = tk.Button(frameC, text = "D").pack()
e = tk.Button(frameC, text = "E").pack()

tk.mainloop()

Another approach could be using the .grid() method

button.grid(row=1,col=0)

the values of row=1,col=0 depend of the position of the other widget in your window

or you can use .place(relx=0.5, rely=0.5, anchor=CENTER)

button.place(relx=0.5, rely=0.5, anchor=CENTER)

Notice that the parameter anchor is referencing the a relative position to the object (in this case button). anchor is not referencing to a position in the window. You could think that the button is a ship that has several anchors so you should choose a coordinate and which anchor you want to fix in that coordinate.

Example using .place():

from tkinter import *  # Use this if use python 3.xx
#from Tkinter import *   # Use this if use python 2.xx
a = Button(text="Center Button")
b = Button(text="Top Left Button")
c = Button(text="Bottom Right Button")

# You can use the strings the referencing the relative position on the button
# strings = n, ne, e, se, s, sw, w, nw, c or center
# Or you can use the constants of tkinter
# N, NE, E, SE, S, SW, W, NW, CENTER
a.place(relx=0.5, rely=0.5, anchor=CENTER)
b.place(relx=0.0, rely=0.0, anchor=NW)
c.place(relx=1.0, rely=1.0, anchor=SE)
mainloop()

You can pass the row and column to the command associated with the function. The two most common ways to do that are with lambda, or with functools.partial.

Here's an example using lambda:

import tkinter as tk

def click(row, col):
    label.configure(text="you clicked row %s column %s" % (row, col))

root = tk.Tk()
for row in range(10):
    for col in range(10):
        button = tk.Button(root, text="%s,%s" % (row, col), 
                           command=lambda row=row, col=col: click(row, col))
        button.grid(row=row, column=col, sticky="nsew")
label = tk.Label(root, text="")
label.grid(row=10, column=0, columnspan=10, sticky="new")

root.grid_rowconfigure(10, weight=1)
root.grid_columnconfigure(10, weight=1)

root.mainloop()