I know ohm's law is probably what i should be using (R=U/I) but I'm missing the value for I on this circuit
You could use 1K or 100K with much the same results. You want to make the input look like a logic one so you connect it to 5V but you want to pull it down with your switch so you don't want a short.
from 0 Ohm (wonder what such a resistance exists..
I thought there was a 30 mA limit on Arduino I/O pins?
5v/10k ohm = 0.5 miliamps