I have fiber to the house. Verizon ran a cat5 wire to my router. The tvs are run off of the same coax as they always have been. The smart TV's and computers and cell phones run wireless off of the router.
You should not have to run the cat 5 unless you want it on the opposite end of the house from where the service company's equipment is mounted. Then you might have to provide the wiring.
Some companies use fiber to the curb and I am not sure what they run from the street to the house.
Either way you will love the quality of the product.