First week I bought my house I fished cat6, cat 5e(phone / alarm), & RG6 to every room in the house.
Pick up a bag of RJ45s, F conns, Decora wallplates & keystones. I’ll loan you the crimp tools.
And… there is no “easy” way to fish the cabling.
Move the router / switch into the basement, and run all your network from a central location, drill lead hole through a floor board, into a stud wall, drill a 1" hole in the drywall where you want the wall plate. Fish tape -or- a wire hanger, duct tape, and your CatX wire. voila. Nice, clean looking install.
…iirc from your thread, you have a split level? If so, it is actually really easy to run cabling to the 2nd level (as with mine), 3rd floor is a bit more challenging depending on the build type.