Translating from english to python, what you want is
while nextNumber != 0:
...keep looping...
So, without changing your code too much, you can simply do:
print ('Please enter your first number')
firstNumber = input()
print('Enter the next number(0 to finish)')
nextNumber=input()
while nextNumber != 0:
if firstNumber<nextNumber:
print ('Up')
if firstNumber>nextNumber:
print ('Down')
if firstNumber==nextNumber:
print ('Same')
# turn the nextNumber into the firstNumber
firstNumber = nextNumber
print('Enter the next number(0 to finish)')
nextNumber=input()
There are also a few things you could improve. For example, you dont need to use the print statement before the input. You can simply do input('Please enter your first number'). Also, you could be using if-else instead of only if.
Here is an example of how it would look like:
firstNumber = input('Please enter your first number')
nextNumber=input('Enter the next number(0 to finish)')
while nextNumber != 0:
if firstNumber<nextNumber:
print ('Up')
elif firstNumber>nextNumber:
print ('Down')
elif firstNumber==nextNumber:
print ('Same')
# turn the nextNumber into the firstNumber
firstNumber = nextNumber
nextNumber=input('Enter the next number(0 to finish)')
Now, you say you want to collect all the Ups and Downs and print them at the end. You can simply create a list and append the value to that list. Like so:
ups_and_downs = []
firstNumber = input('Please enter your first number')
nextNumber=input('Enter the next number(0 to finish)')
while nextNumber != 0:
if firstNumber<nextNumber:
ups_and_downs.append('Up')
print ('Up')
elif firstNumber>nextNumber:
ups_and_downs.append('Down')
print ('Down')
elif firstNumber==nextNumber:
ups_and_downs.append('Same')
print ('Same')
# turn the nextNumber into the firstNumber
firstNumber = nextNumber
nextNumber=input('Enter the next number(0 to finish)')
print(ups_and_downs)