You can go about this a few different ways, depending on how much wire you want to run and how close the devices are to each other.
It is unclear from your diagram how many floors we are talking. Generally you want your wireless router or access point to be either centrally located or where you will be using it most.
I'm more a fan of equipment over wiring. I'd recommend one network connection per room at a minimum, one on each wall is better and allows for more flexibility.
Since it looks like your modem is going to be in your basement, I would recommend running all of your cable to the basement.
Get a Wall Mount cabinet you probably won't need more than a 6U, but since those are out of stock until September you may want to go with a 9U now. Then you're going to need a patch panel with sufficient ports for all of your runs, jacks one for each run, wall plates to install the jacks in, cable to connect the jacks with the patch panel.
Now for equipment. You're going to want a router by the modem. In rooms with a lot of devices you may want only one run with a switch to connect all of the devices rather than one run per device. I'd recommend going with ZyXEL routers they are inexpensive and you can easily configure them as wireless access points. Routers configured as access points will have all but one LAN port available to be used as a switch.
Your basic layout will look like this:
Modem -> Router -> Patch Panel -> Device/Switch/Access Point