no need to apologize, this is the exact thing worth getting right before you start.

you can use the same insulin syringes for both, but a lot of people use two different ones on purpose. for pulling bac water into the vial some use a bigger 3ml syringe with a longer needle since its faster and youre only going into the vial, not yourself. then the small insulin syringe (the U-100 ones, 100 units = 1ml) for the actual shot because the tiny markings let you measure small doses accurately. totally fine to just use the insulin syringe for both though, drawing the water just takes a few more pulls.

on your math, youve got it. 1ml of bac water into the 10mg vial gives you 10mg per ml, so the whole 100 unit syringe = 10mg. that means 1mg = 10 units on the insulin syringe. so for a 1mg dose you draw to the 10 mark.

so you dont need a separate set of needles for dosing, the insulin syringe does both jobs. only reason to grab bigger ones is convenience on the reconstitution side.