There's several possible ways to approach this. The best one is going to depend on your desires.
If you're happy with the existing wifi performance with the wifi router on the main floor of the house, and you want that device to continue being your main router (firewall, NAT, DHCP, DNS, etc.) you can leave it there, but would require pulling a second Ethernet cable between it and the basement (parallel to the old TV cable you're currently leveraging).
In that case, in the basement, you'll have the cable line coming in the house --> the cable modem --> WAN port of the router upstairs (likely using the second Ethernet run you'll need). Then LAN port 1 of the router would leverage the old TV Ethernet cable run back downstairs connected to port 1 of an
8 port Gigabit switch. Port 2 of the switch would connect to a single Ethernet cable running to your garage, as shown in your diagram. Ports 3-8 can be used to connect all of your 10/100/1000 Mbps wired PCs/devices in the house (as well as LAN 2-4 on the router upstairs, if needed). Then, in the garage, you have a couple options. The cheapest method would be to install a wifi router there that's capable of running in AP mode, with the Ethernet cable that's coming from the house connected to LAN port 1 of the wifi router. Your other wired devices in the garage would plug into LAN 2-4 of the router; don't use the WAN port. You would need to ensure that the wifi router you place in the garage is capable of running in AP mode (check the manual), or you need to buy a router that's compatible with aftermarket firmware like Tomato, DD-WRT, Gargoyle, etc. You'd then simple setup the wireless (SSID, encryption, password, etc.) the same as you have it in the house. The other option in the garage is to land the Ethernet cable from the house onto port 1 of another 4/8 port switch, then also buy/install/connect an AP there. And a third option in the garage is to simply place any wifi router there, with the home ethernet cable connect to the WAN port, and configuring the wifi as a different wireless network (different SSID). This third option is easiest to install, but not as nice to manage or maintain... and requires 2 SSIDs. You'd also need to ensure you use non-overlapping IP ranges between the networks.
Now, if you really don't want to run the second Ethernet cable between the basement and router upstairs (old TV run), then you'd need to move the router back downstairs (disabling wireless, and replacing the switch), and place a new AP upstairs in the same spot as the current router. That AP could be a purpose built AP, or a re-purposed wifi router running aftermarket firmware in AP mode.
In all scenarios, make sure you're pulling CAT6 cabling especially on the house-garage run.
You could pull 2 cables between the house & garage if you really needed the Gigabit bandwidth for each connected device, but I don't believe that's the case here. A single Gigabit uplink should be plenty for connecting all devices in the garage to the main house network.
You could also use a Powerline adapter instead of pulling a dedicated CAT6 cable between the house & garage, but then your uplink will likely be limited to 100Mbps or less vs. 1000 Mbps. You're also going to be putting "noise" on your electrical lines, which usually isn't a problem, but I prefer not to due to HT equipment as well as not trusting DTE's power pole config to isolate my network from my neighbors. But I'm more paranoid than most.
Assuming that the garage is detached, if you were already planning to run a trench (say for a new electrical sub panel), I'd certainly pull 1-2 CAT6 Ethernet cables at the same time. But I wouldn't trench just for the Ethernet - several other possible solutions.
Big decisions I see you needing to make: 1) Can/are you going to be able to easily run Ethernet between the house & garage? 2) Are you willing to run a second Ethernet cable between the basement and where the router currently sits?