My house was built in 1947 and has old wiring. I want to eventually rewire the whole upstairs, but I am concentrating on the bathroom right now. It looks like, the easiest way to do it is to bring 1 wire from the basement to the attic, and branch out from there to each room. My guess is about 15 ft from the basement to the attic. I think the easiest way is to bring 1 wire up, but I want to know if this is the best way.

I will try to move this post to electrical because where you have it now no electricians will see it. We would need to know a lot more about your home to give good advice and, most likey, this work needs a permit and is not a diy project. Your current line of thinking would be putting every bedroom light and receptacle on one breaker, not going to work.

Yes, if you're doing a home upgrade project, do it in accordance with the current code...
Get permits and inspections, protect your family and your investment.

The size of the cable you need is dependent on the Amperage selected for the circuit.

For example, a 15 amp circuit require a #14 AWG cable (or larger) while a 20 amp circuit requires a #12 AWG conductors (or larger).

What is the amperage you need on the circuit?

Be safe, 1 circuit per bedroom. If you can pull 1 you can pull 3. They will need to have AFCI breakers too.