I'm an instructor, and I happen to have a very large urban, left leaning social circle, so I have a lot of opportunities to introduce new people to shooting. In fact, my father is trying to convince his wife to agree to buy a gun for their home, so I took the two of them to the range this weekend.
One thing to keep in mind when you're working with new shooters is that you need to keep it simple. They don't need to know the name of every part of the gun, or the advantages of .45 over 9mm, or proper breathing techniques while shooting. You just need to be sure they're safe, comfortable, and hitting the paper under 10 yards.
Before we step foot on the range or even see a gun, I spend at least a solid half hour with a new shooter going over safety. I go over the first four Rules, first reciting the rule, then explaining and discussing each one with some examples or anecdotes, answering any questions, then having them recite the rule back to me. Sometimes I'll use a blue gun for demonstration, but I'll keep the firearms stowed (tends to distract at this point in time). Then I'll talk a little about what I call "range etiquette." We'll talk about eyes and ears, what to do if the gun jams, the Cease Fire command, etc, etc.
I also have a handout with all this info (bulletin points, no details) that I hand out.
Then we'll go to the range, and I'll walk through the basic operations of a .22 revolver (Smith 617) - how to load, grip, single action & double action trigger pull, and a very basic how-to-shoot (sight picture, sight alignment and a word or two on trigger squeeze).
I'll load and fire a couple rounds with them watching, so they know what to expect when the gun discharges. Then I'll have them dry fire a couple times. Then they get to shoot, usually 2-3 cylinders. I never use a target, just a blank piece of paper.
Then we move onto a semi auto .22 (Ruger MkII), and we basically follow the same steps. Then, if they're doing well and comfortable, I'll move them up to .38 Special (Smith 686) and 9mm (Beretta 92FS), using the same procedure.
From there, the next step depends entirely on WHY I have the person at the range, and try to personalize the experience - educational, fun, etc. Maybe we'll move onto long arms, maybe we'll try out more of a variety of small and medium caliber handguns, maybe we'll move onto big stuff. They key is always focusing on safety, comfort and simplicity.