My husband uses that book, mangeorge. It’s excellent. Full color, very easy to understand photographs. He’s rewired about 75% of our house (we had aluminum wiring instead of copper). He’s installed 3-way dimmer switches, ceiling fans, lights, and redone the outlets in most of the house.
FWIW I had the EXACT same problem about a year ago. As I recall, the new switch contained instructions explaining what to do. So take a look at the instructions.
My recollection is that, in essence, I wrapped the switch in electrical tape and left it ungrounded.
Built 1966… then do your outlets have a ground?
I had a house built in 1966 that had grounded outlets but I owned one house that was built in 1976 that didn’t have grounded outlets or any ground wires.
Grounds are green wires.
Your Question: “Where do I put the green/ground from the new switch?”
If your outlets are not grounded you have no ground.
Then just forget about the green ground wire.
FYI
The grounds can either be green wires or just bare exposed copper too.
I just noticed The Big Cheese lives in Madison, and whuckfistle lives in Milwaukee. Is this a long drive? If not, perhaps whuckfistle could drive over to The Big Cheese’s place and help him out. In exchange, The Big Cheese could buy the beer and grub. 
Hour and a half away or so, Crafter_man I do have these Leinies sitting in my fridge…Nah, I can handle it, or back out gracefully.
**Mangeorge,**I’ll check out the book, thanks. I’ve got a ceiling fan to install next.
Lucwarm, the new switch I bought assumes you are replacing one that has the same amt & types of wires, or you know what to do if it’s different. I’ve got this extra green/ground one on the new switch. I’m guessing I could leave it ungrounded as someone suggested, but now I’m curious as to what the right thing to do is. Of course the right thing seems to be a little above me. Seeing someone do it I could probably get it, doing it from text is another thing. I hope I’m not sounding ungrateful.
**Olefin,**That was one of my prior questions. Since it solely has 2 hots coming out of it, there probably isn’t a ground?
I haven’t been shocked yet…of course it only takes one time 
You are correct… my error.
I was thinking most people had the common sense to know if the wire had no insulation it had to be the ground.
My opinion, if they don’t know that they shouldn’t be messing with the wiring… they should call an electrician.
I`m a Bud man, and yes, I realize I live in Miller Town. [said as I shield myself from the glares of fellow Milwaukeans.]
“the old light switch, when pulled out, has a black wire coming from the back/top and a red wire coming from the back/bottom. That’s it-2 wires.”
Take the old one to the hardware store & ask them to give you a dimmer & instruct you on how to wire it. I always take the old parts to the store.
Ok, TBC, so you’re gonna install a ceiling fan next. I know you can do it, but here’s where you really want to be careful. Follow instructions exactly, especially the part about the special box. It’s heavy-duty and braced to withstand the extra weight and torsion load of the fan. My son-in-law didn’t, and almost wound up wearing the fan. Lucky for him and my daughter the wires held. Wish I was there.
Anyway;
Get a book. The one I recommended is a good one.
Install the switch. You can do this first if you want.
Ask CIL how you did when he arrives. He’ll shrug and say “ok, I guess”.
Feel all confident.
Install fan, exactly as instructed.
Feel even more confident, and spend the money you saved on something really cool. Like maybe a nice dinner for your SO and yourself. 
As far as ceiling fans go…
The more they cost the better they are (generally of course). The high end ones are quieter and balanced so they won`t hum or wobble.
Trying to balance a cheap fan is a bitch.
And DO make sure the ceiling box is rated for fans as mangeorge stated.