OP, as others have stated - the boxes will just use co-ax, nothing else, even with whole-home. They talk to each other via MoCA on top of the co-ax.
TW won't touch your Blu-Ray players or desktops in separate rooms. I'd call an electrician, home theater wiring contractor (it's low voltage so you don't technically need an electrician), or buy the tools and run the cable yourself if you want to be cheap.
Cheap is relative though, because decent tools are going to cost a fair amount, and experienced contractors are going to be better at fishing wire than you will be. I wired my own place (and bought the tools), but I'm an IT guy and already know how to terminate Cat5e.
If you run wire, run way more than you think you need, or at least have them use conduit so you can add more later. (I'm kicking myself for not using conduit because there are a few spots I want more wire than I originally ran.) I would run at LEAST 2 ethernet to each location, preferably 3 or 4 in areas with a TV / etc (game systems, Roku, all that stuff uses ethernet). You can always drop in a little switch if you run out of ports, but then you have another wall wart to deal with and the potential for bandwidth starvation if you are trying to do something really network-heavy in the house (this is unlikely if you're just gaming / streaming internet video though).
For coax, I'd do at least 2 runs of that as well. Cable only uses one, and DirecTV is now down to single-wire (with SWiM), but I believe Dish may still require a pair in some instances. It's also cleaner if you decide you want two DVRs in one location - two home-runs instead of a split behind the TV.
Also, don't spend extra money for CAT6. CAT5e is sufficient for gigabit ethernet. CAT6 gains you nothing and is harder to work with and more expensive. If you really want to be ready for 10Gb networking in the house, you want CAT6A
(augmented) or CAT7, both of which are very pricy and a pain to terminate. Normal CAT6 doesn't have enough alien crosstalk rejection to run 10Gb more than about 50-55 meters, while C6A or C7 are both good for the full 100 meter Ethernet spec.