For running up the second story, if it is a stick frame house, you have a couple of options. If you have central AC, you can run the wire from your current router through the wall, or even along the baseboard in to an AC duct and then snake up through the duct work up to the second floor. You'll want/need to use plenum rated wiring as it is running through your HVAC duct work.
Alternately, you can just open the wall and run the wire up to the next floor. It helps if you have walls between floors that are placed on top of each other, but you can always run the wiring up through the header of the wall on the first floor, down the joists to below a wall on the second floor, up through the footer of that wall and in to a wall box and stick a keystone jack in it, or just up through the baseboard there.
You won't have to cut too many holes, so long as there is no blocking in the wall. Just one to have it enter, then at the top of the wall so you can drill the header and then upstairs a hole in the wall to go down through the footer to pull the wire up. A wire fish would be useful for all of this.