This link explains how to print to the screen in a MIPS simulator like QTSPIM or MARS.
As for the code, there were a few bugs. The line li $a0, 0 is overwriting the work done by the initial la $a0, Array instruction because the li is setting the base address of your array to 0. Instead, you should move the la instruction into the loop so that $a0 is properly reset to the base address of Array after iterating over the entire array, and remove the li instruction. You also will need to add in a condition for when your program has completed the sort. I would suggest the following revisions (tested with SPIM):
main:
la $t0, Array # Copy the base address of your array into $t1
add $t0, $t0, 40 # 4 bytes per int * 10 ints = 40 bytes
outterLoop: # Used to determine when we are done iterating over the Array
add $t1, $0, $0 # $t1 holds a flag to determine when the list is sorted
la $a0, Array # Set $a0 to the base address of the Array
innerLoop: # The inner loop will iterate over the Array checking if a swap is needed
lw $t2, 0($a0) # sets $t0 to the current element in array
lw $t3, 4($a0) # sets $t1 to the next element in array
slt $t5, $t2, $t3 # $t5 = 1 if $t0 < $t1
beq $t5, $0, continue # if $t5 = 1, then swap them
add $t1, $0, 1 # if we need to swap, we need to check the list again
sw $t2, 4($a0) # store the greater numbers contents in the higher position in array (swap)
sw $t3, 0($a0) # store the lesser numbers contents in the lower position in array (swap)
continue:
addi $a0, $a0, 4 # advance the array to start at the next location from last time
bne $a0, $t0, innerLoop # If $a0 != the end of Array, jump back to innerLoop
bne $t1, $0, outterLoop # $t1 = 1, another pass is needed, jump back to outterLoop
Be sure to check out this link for additional examples and explanations on what each MIPS instruction does.