How to get objects move in a pygame program?

There are multiple (many) problems with your variable names, assignments and the logic.

The core of the problems are in the movement code. a Box is a surface box1, and has co-ordinates box1X and box1Y. The boolean moveBox1 determines if the keyboard input is for the selected box (keys: 1,2,3,4).

The code governing movement left is:

elif event.key == pygame.K_LEFT :
    if moveBox1==True and box1X>=30:   # <-- this clause is OK
        box1X=box1X-30
    elif box2==True and box2X>=30:     # <-- WRONG, not "box2", use "moveBox2=="
        box2=box2X-30                  # <-- WRONG, not "box2", use "box2X="
    elif box3==True and box3X>=30:
        box3=box3X-30
    elif box4==True and box4X>=30:
        box4=box4X-30

So if the user was trying to move box2, it's not possible, and is corrupting the surface stored in box2 with the number of box2X-30. Similarly there's issues for all the other movement code.

In the K_RIGHT movement clause:

box2X = box2X + 540

Which moves the box to the other side of the screen, is this intended?

I patched the code to the point where I could move box1 left & right by 10 pixels at a time:

import pygame

pygame.init()
screen = pygame.display.set_mode( (640, 480) )
pygame.display.set_caption("Wahoo")


background = pygame.Surface(screen.get_size())
background = background.convert()
background.fill( (204,204,255) )

# Not supplied by OP
def labelBox( surface, colour, label ):
    surface.fill( colour )

box1 = pygame.Surface((100,100))
#box1 = box1.convert()
labelBox(box1, (153, 255, 255), "1")
box1X = 0          # The starting location for this box
box1Y = 0
moveBox1 = False   # If True, this is the box that moves

box2 = pygame.Surface((100,100))
box2 = box2.convert()
labelBox(box2, (255, 255,153), "2")
box2X = 540        # The starting location for this box
box2Y = 0
moveBox2 = False   # If True, this is the box that moves

box3 = pygame.Surface((100,100))
box3 = box3.convert()
labelBox(box3, (153, 255, 153), "3")
box3X = 540        # The starting location for this box
box3Y = 380
moveBox3 = False   # If True, this is the box that moves

box4 = pygame.Surface((100,100))
box4 = box4.convert()
labelBox(box4, (255, 153, 204), "4")
box4X = 0        # The starting location for this box
box4Y = 380
moveBox4 = False   # If True, this is the box that moves



clock = pygame.time.Clock()
keepGoing = True

while ( keepGoing == True ):

    clock.tick(30)   # Maintain 30 frame per second refresh.


    for event in pygame.event.get() :
        if event.type == pygame.QUIT :
            keepGoing = False
        elif event.type == pygame.MOUSEBUTTONDOWN :
            box1X=0
            box1Y=0

            box2X=540
            box2Y=0

            box3X=540
            box3Y=380

            box4X=0
            box4Y=380

        elif event.type == pygame.KEYDOWN :
            print("KEY...")
            if event.key == pygame.K_ESCAPE :
                keepGoing = False

            elif event.key == pygame.K_1 :
                moveBox1=True
                moveBox2=False
                moveBox3=False
                moveBox4=False

            elif event.key == pygame.K_2 :
                moveBox2=True
                moveBox3=False
                moveBox4=False
                moveBox1=False

            elif event.key == pygame.K_3 :
                moveBox3=True
                moveBox4=False
                moveBox2=False
                moveBox1=False

            elif event.key == pygame.K_4:
                moveBox4=True
                moveBox1=False
                moveBox2=False
                moveBox3=False

            elif event.key == pygame.K_LEFT :
                print("LEFT")
                if moveBox1==True and box1X>=30:
                    box1X -= 10
                    print("MOVE 1 LEFT, box1X=%d" % (box1X))
                elif moveBox2==True and box2X>=30:
                    box2X -= 10
                elif moveBox3==True and box3X>=30:
                    box3X -= 10
                elif moveBox4==True and box4X>=30:
                    box4X -= 10

            elif event.key == pygame.K_RIGHT :
                print("RIGHT")
                if moveBox1==True and box1X<=540:
                    box1X += 10
                    print("MOVE 1 RIGHT, box1X=%d" % (box1X))
                elif moveBox2==True and box2X<=540:
                    box2X += 10
                elif moveBox3==True and box3X<=540:
                    box3X += 10
                elif moveBox4==True and box4X<=540:
                    box4X += 10

            elif event.key == pygame.K_UP :
                if moveBox3==True and box3Y<=580:
                    box3Y==box3Y+580
                elif moveBox4==True and box4Y>=580:
                    box4Y=box4Y+580
                elif moveBox2==True and box2Y>=580:
                    box2Y=box2Y+580
                elif moveBox1==True and box1Y>=580:
                    box1Y=box1Y+580

            elif event.key == pygame.K_DOWN :
                if moveBox4==True and box4Y>=380:
                    box4Y=box4Y-380
                elif moveBox1==True and box1Y>=580:
                    box1Y=box1Y-380
                elif moveBox2==True and box2Y>=580:
                    box2Y=box2Y-380
                elif moveBox3==True and box3Y>=580:
                    box3Y=box3Y-380 

    screen.blit(background, (0,0))
    screen.blit(box1, (box1X, box1Y))
    screen.blit(box2, (box2X, box2Y))
    screen.blit(box3, (box3X, box3Y))
    screen.blit(box4, (box4X, box4Y))

    pygame.display.flip()

I don't want to sound harsh - because we all beginners once, but this code is a good example of why data structures and classes, although more work initially, make the coding much easier and faster.

The OP's boxes could easily stored in a list, like:

box1 = [ "Box1", Surface(100,100), (0,0),   True  ]
box2 = [ "Box2", Surface(100,100), (500,0), False ]
...

box_list = [ box1, box2, ... ] # for N boxes

And then just iterate through the list, doing the processing:

#... LEFT key pressed
for box in box_list:
    label, surface, coord, selected = box
    if ( selected == True ):
        ... do stuff to move box

This removes the tyranny of having a huge bunch of similarly-named position variables. With so many, it's easy to get confused with all their comings & goings.

So, better yet, put everything about a Box in a class, which keeps it all together, and would provide more meaningful "readable" code:

class Box:
    def __init__( self, label, size, position, colour ):
        self.surface  = pygame.Surface( ( size, size ) )
        self.rect     = self.surface.get_rect()
        self.rect.x   = position[0]
        self.rect.y   = position[1]
        self.selected = False
        self.surface.fill( colour )
        # TODO: put label on box

   def moveLeft( self ):
       self.rect.x -= 30

   def paint( self, window ):
       window.blit( self.surface, ( self.rect.x, self.rect.y ) )

...

#... LEFT key pressed
for box in box_list:
    if ( box.selected == True ):
        box.moveLeft()