It depends where you live. You could call Cox to verify if fiber-optic cable is available at your address, or you could use a third-party website, such as https://broadbandnow.com/.
I don't know how updated it is but after entering your zip, there will be a list of providers in your area. Pay attention to the li'l circled icons for Cable (Coax), DSL, Fiber, Satellite.
If fiber is available to you, Cox will run fiber-optic cable to your house, install a Network Interface Device (NID) outside your house and an Optical Network Terminal (ONT) inside your house.
If fiber is not available, your only other option is coax but that'll be a short run from the NID to a wall-plate in your living room. This wall-plate will be your Primary Connection Point. It reads like the holes are already drilled so a tech only needs to run cable and terminate. The latest iteration of coax cable for Internet and cable-TV is RG-6, so hopefully the cable you dumped was the antiquated RG-59.
As far as the rest of your house...
If you don't want cable-boxes in other rooms, you won't need to run coax.
If you don't want landline telephones in the other rooms, you won't need to run telephone line (RJ-11).
If your wireless coverage stinks and you feel like running cable, run some Ethernet!
