Sorry for the beginner question. I'm working on a nixie tube clock and I have a bunch of shift registers for each tube. Do I need to use a different clock for each IC or can I use the same clock for each register?
Assuming you're using something similar to a 74HC595 shift register, then, yes, you can wire the same clock to all registers.
There's a limit, but it is far more than the 6 shift registers you want to use, so you can safely wire the 6 clocks together without a problem.
You can also wire the latch pin of all of them together.
If the device you need to control has 10 inputs, then you must string more shift registers together and allocate the first register to the first 8 inputs, and the remaining 2 to the second and so on.
So, calculate how many inputs you'll have in total, and then string together the corresponding number of shift registers.
Does that make sense to you?