f1 = Mod[#, 4] &;
f2 = Mod[#, 7] &;

SortBy[Range@10, {f1, f2}]

{#, f1@#, f2@#} & /@ % // Grid

{8, 4, 1, 9, 5, 2, 10, 6, 7, 3}

string = {"cat", "fish", "catfish", "Cat"};
Sort[string]
out: {"cat", "Cat", "catfish", "fish"}

ToCharacterCode[string]
out: {{99, 97, 116}, {102, 105, 115, 104}, {99, 97, 116, 102, 105, 115,104}, {67, 97, 116}}

SortBy[ToCharacterCode[string], First]
out: {{67, 97, 116}, {99, 97, 116}, {99, 97, 116, 102, 105, 115, 104}, {102, 105, 115, 104}}

FromCharacterCode[
    SortBy[ToCharacterCode[string], First]
]
out: {"Cat", "cat", "catfish", "fish"}