Answers mixed in below:
So is the 1 pin of the 7-segment display the positive and the 5 the negative or vice versa?
Or are both positive? I think it's that, right?
Both pins 1 & 5 go to the positive of the supply. They are internally connected on the device.
So then the wiring for the rest of the 7-segment is, for example,
DRAIN0 (on shift register)-33 OHM Resistor-Pin 7 on 7-segment display
Yes.
(The batteries I'm using actually are 1,500 mAh)
If you have all 7 segments of a digit on, that's 7 x 20mA = 140mA, I don't remember how many digits you have altogether. 4? Then if you all 4 displaying "8888" you can expext to draw 560mA, you will get maybe 3 hours of life form 1500mA batteries. Most likely longer as you will have fewer segments on at any one time.
I have a common ground to the shift register (just to one of the GND pins, right?), battery pack, and Arduino.
Connect all 3 grounds from the shift register.
So the SER IN is the shift_data_out line and SER OUT goes to the next SER IN.
Yes.
I assume that the pins of the Arduino for the shift_data_out line, shift_out_clock line, and the load_data line don't matter?
Correct.
I think I'm just a bit confused about those 3 lines in general.
I believe I understand the shift_data_out line.
What pins on the shift register does the shift_out_clock line and load_data line go to?
shift_out_clock goes to SRCK
load_data goes to RCK
Low on SRCLR will clear the input register, tie it high if not used.
Low on G will enable the output transistors so 0s & 1s will show up on your display. With G high, no outputs will turn on.