Jim, I don't think you realise how much we admire you. Age, health and, no doubt, other issues. Bravo.
Now, as this is, in your own words,
a brand new Mac:
Connect the Mac to the mains via whatever cable came with it so that you are not dependent on the battery.
Start up the Mac. Then, as Jake has so patiently said, let it go through the introductory stages eg choose your language, time zone and all the rest. At some point, it will ask for your (actually your wife's) Apple ID - username and password.
As you plough through, there will come a time when you are asked if you want to transfer your data/documents/setting etc from - choose Time Machine. And that will transfer everything (except, obviously the previous Operating System) such that your new Mac will operate as your older one did - but with a new Operating System.
You may need to go into System Preferences > Software update to make sure that you have the latest version of Monterey, which is 12.5.1
Your Admin login to the Mac will be the same as before, and your settings likewise.
Please don't feel embarrassed to ask any question you like. We're with you
Ian