So I’m teaching my kid how to drive a stick shift. I try to give him the lowdown on the differences between automatic transmissions vs. manual transmissions. What I want to know is, when did automatics become widely available? I would guess sometime during the 1950s. But which manufacturer mass produced them, and on what car?
Power-Glide. GM. I’m guessing about 1955. Available on GM products from then forward. Did the '54 'vette have an auto tranny option?
The Ford Model T was a semi-automatic. You had to manually select the gear; but there was no clutch. It used a planetary gear like a modern automatic transmission.
When Ford dropped the Model T and went to the Model A, they used a manual transmission.
The Car History Timeline (by month) notes that Oldsmobile introduced a column-shift automatic in 1937 and that Chevy introduced the Power-Glide in 1950, “first low priced car to do so.”
This would match my memory as we had a 1952 Chevy with an automatic.
http://members.shaw.ca/bcoldsclub/milestones/milestones.html
Clearly the AT was a 1950’s marketing bonanza the car companies aimed at suburban Dad-Car family units. Convince Dad that his last-year-car with that monstrous clutch and shift thingy is barbaric - and that even he would appreciate driving a new classy Olds with a Superb-Matic transmission. “Look!” says the salesman, pointing to some faux-chrome lettering on the front quarter panel, "this limited edition vehicle has Superb-Matic!
If you’re teaching your kid to drive, Hermann C - manual is the best. Acceleration and gas mileage are superior. It’s an extension of bicycle riding, the mechanics are elegant and represent how cars work. (subtle) Both feet and both hands are used. 
Having driven the Model T I don’t recall any “manual” gear selection.
You had to be pretty good with your feet-----that’s how you accomplished forward and reverse-------and also panic stops.
Manual spark and fuel control------but not gears.
In the early 30s one could buy a driveable T at a junk yard for about 6 bucks fill it up for about another buck and drive it away
You raised hell with until you killed it
Then you took of the plates and left it where it died.
And both hands and feet get tired and sore when you are forced to drive and hour each day in rush hour traffic! Honestly I do not see the appeal of manual transmissions. They require 4 times the effort for a marginal gain in performance. I’d rather have the convenience and faster reaction times of an automatic.
The three speed automatic is nothing but a Model T transmission with a hydraulic shift control. A torque converter replaces the clutch.
For first gear a brake band stops one end of the planetary gear train. For second gear another brake stops the other end. For reverse both ends are stopped. Third gear is with both ends free and the planetary gear set rotates as a unit giving direct drive. In the T the brake bands were operated by foot pedals with reverse being a lever. In an automatic a hydraulic system works the brake bands.
To all who answered on the automatic transmission question.
I don’t know what model of the T you guys drove but the left pedal was the clutch.
Push it all the way down and you were in “low”
Get rolling and release the pedal and you were in"high
To stop you pushed the left pedal down half-way to the neutral point ands applied the brakes
To reverse you pushed the left pedal half-way down to neutral and then pressed the middle[reverse] pedal.
I don’t remember any forward/reverse lever.
What model of the T had such a lever?
Or maybe my octogenarian memory has a slipping clutch
I would disagree (with the first part anyway). You should start out with an automatic. Driving is pretty intense the first few times and the extra added skill that a manual requires doesn’t help.
I do, however, absolutely feel that soon after getting the hang of everything else you be taught how to drive a stick. It isn’t that hard. Whenever I come across someone over 18 who says, “Oh, I can’t drive a stick” I literally laugh in their face and ask if they can walk & chew gum! 
This site about the Model T transmission says that your memory is better than mine.
One of my uncles had a 1920something Dodge touring car. Maybe that was the one with the reverse lever.