I recently did this to my new flat (apartment) (although mine was a retrofit to an existing building which was a fair bit more difficult than installing it in a new built) - I made a YouTube video documenting the installation process:
https://youtu.be/uGRYfVdAja4 and showing the finished result:
https://youtu.be/5p1s5XSjBP8. The videos are obviously fairly specific to my installation although the general hardware used and layout should apply elsewhere.
Here's an album of my adventures ripping holes in my walls/ceilings to install the cabling:
https://photos.app.goo.gl/mAHnuspTJYNojU1T9Generally what you'd want to do is run all of your cables back to a central point in the building and terminate them at a patch panel, this central point is where you would have your network switch and other distribution hardware. For networking you want to run standard twisted pair cable, for most situations I'd go with CAT6, higher ratings such as CAT6A would be worthwhile only if you plan on running 10gbit networking over it however this cable is much more expensive, bulkier and harder to run so I tend to stick with CAT6 which can still technically do 10gbit over shorter lengths. The only bit of advice I'd give is to make sure that whatever cable you buy is pure copper, some unscrupulous sellers (particularly the extremely cheap cable on eBay and Amazon) use copper clad aluminium which is a much cheaper, lower quality cable.
I'd recommend dropping multiple CAT6 cables to each room in sensible locations near to where devices may be located. I'd always put in more jacks than you need as it's relatively cheap to do and it's better having ones you don't use rather than lacking ones that you do. For example, you'd probably want 2-4 ports behind each TV to allow for smart TVs, set top boxes and games consoles. I'd also run cables for use with wireless access points in sensible, central locations. In my case I mounted the access point on the ceiling in the hallway.
The CAT6 cable can also be used for things other than networking, with special hardware you can send HDMI video signals over it which could be useful. It's also suitable for use with analogue telephones so don't bother running those separately, just run more CAT6 as it's more useful. Then, at the central location, rather than connecting the telephone ports to a network switch, you would link them up to your incoming phone line.
As for coax, I'm not familiar with the Cable TV networks in the US although you're probably going to want to run at least one to each location where you plan on having a TV. I'd forget about using the coax for any sort of networking, the hardware that exists to do this is expensive and only really designed for use in environments where coax is already in place and the user doesn't want to run new cables, there's no reason to use them where installing actual CAT6 is an option.
