How can I sort multiple Python lists based on the sort applied to one list?

You can sort a list of indexes from a range, based on the values in one of your lists. Then you can use the indexes reorder the original three lists:

store = ['Store number', 2, 3, 4, 1, 5]
var1 = ['Variable 1', 82, 99, 44, 32, 65]
str1 = ['String 1', 'cat', 'dog', 'lizard', 'alligator', 'crocodile']

sort_indexes = sorted(range(1, len(store)), key=lambda i:store[i])

store[1:] = [store[i] for i in sort_indexes]
var1[1:] = [var1[i] for i in sort_indexes]
str1[1:] = [str1[i] for i in sort_indexes]

Your data structure would be a lot nicer to work with if the header for each column wasn't contained in the same list. But that awkwardness doesn't actually effect this code much, since we're not directly sorting the lists anyway, we're sorting indexes instead, and there's not really any problem doing that starting at index 1 instead of 0. If you move the headers out of the data, you'd just be able to skip the slice assignment at the end of this code (just use the list comprehension values directly).

If you must use your current data structure, here is a function that will do what you want. Note there are probably much more clever approaches, but this will work.

given:

lolists = [['Store number', 2, 3, 4, 1, 5],
            ['Variable 1', 82, 99, 44, 32, 65],
            ['String 1', 'cat', 'dog', 'lizard', 'alligator', 'crocodile']      
            ]
def sortinnerLists(lol):
    kys = lol[0][1:]
    sortedKys = sorted(kys)
    rslt = [[lol[0][0]], [lol[1][0]], [lol[2][0]]]
    for v in sortedKys:
        indx = kys.index(v)+1
        rslt[0].append(v)
        rslt[1].append(lol[1][indx])
        rslt[2].append(lol[2][indx])
    return rslt    
sortinnerLists(lolists)  

yields:

[['Store number', 1, 2, 3, 4, 5],
 ['Variable 1', 32, 82, 99, 44, 65],
 ['String 1', 'alligator', 'cat', 'dog', 'lizard', 'crocodile']]