@cavaughan Once you know addition and subtraction, then multiplication is just repeated addition, with shifts to the left for each digit in the multiplier. Division is similar, but starts at the left of the dividend, and you subtract repeatedly until it underflows (the machine will ring a bell!) and then you add one back and shift right. Repeat until you have enough digits of precision to suit you.
@cavaughan I have a Comptometer manual that goes into how to do more advanced things like roots, but it seemed complicated enough that I'd never know if I got the right answer.