Can't get 4 digit 7 segment display to show different values
You have a common cathode display and all your cathodes are tied to ground through resistors. You have no way at all of separating out the different digits.

You cannot directly drive such a display with a shift register like that - you need a more direct connection to the Arduino (it needs to be high speed) and rapidly switch each cathode on and off in sequence to display each of the four numbers one after another.

Also the resistors should be in the 8 anodes, not the 4 cathodes.