I agree with the others, start with the rifle.
I teach NRA Basic Pistol and Home Firearms Safety classes. But in addition, I've informally introduced a number of people to firearms. When I teach someone informally, we start at my home, where we spend about an hour going over the basics of firearms safety, manipulation, and marksmanship. By starting at your home with empty guns, you can teach her in a more teachable environment -- it'll be a lot quieter than the range, have a lot fewer distractions, and she'll be a lot less scared.
Go over the big 3 safety rules (1. safe direction, 2. finger off the trigger, 3. action open and unloaded until ready to use) a number of times. The purpose of the repitition is two-fold: first, the repitition helps your student learn the rules and second, the emphasis you put on safety helps calm their fears (and sets a good example). They are afraid, and for good reason -- if they mishandle the gun, then they or someone else could be injured or even killed. So part of your job is convey to your students that the <b>most important thing to you is safety </b>(and if it isn't, then you shouldn't be teaching...). You should be able to tell your students that you are a safety nazi and you are proud of it.
I suggest that you start the first time with just the rifle. It's the easiest to hit with and has the least recoil. While still at home, demonstrate the positions that you are going to use at the range (seated, prone, bench-rest, etc.). Have her try those positions (with an empty gun, of course). Have her get comfortable holding and manipulating the gun (safety on off, bolt open and closed, etc.) in the safety and quiet of your own home.
While at home, tell her about the basic rules of the range you are going to (e.g., no handling of firearms while someone is down range, anyone can yell "cease-fire", what to do if someone yells "cease-fire", etc.).
When you do get to the range, first reiterate the big 3 safety rules. You cannot repeat these too often.
You should demonstrate each position again and fire the first shots. Show her how it is done. She doesn't know how loud a gun will be or how much it will kick. When you take the first shots, you show her what to expect.
She'll likely be quite nervous (note, I'm not being sexist here -- almost all of my first time shooters, whether male or female, are very nervous at first). That means you have to be very calm and supportive. When she does take her first shots, you should be gently talking her through each step -- breath, relax, align the sights, get the proper sight picture, focus on the front sight, squeeze the trigger. Use a nice steady, even tone, focusing on being a calm, steady, reassuring presence right next to her. Correct mistakes gently (unless they are going to cause an immediate safety issue, of course). Celebrate her successes.
Once she starts to get the hang of it, don't over teach -- stay right by her side, but reduce the amount of instructions that you are giving her. Some mistakes are better left uncorrected. For example, if she ejects the last round, but doesn't realize it is the last round, closes the bolt and starts to shoot, let her continue. She'll find out in a second that the gun isn't loaded when it dry-fires, and you'll get a chance to see if she's anticipating the recoil.
Take advantage of teachable moments. For example, if she has a misfire, explain to her that 9 times out of 10, a misfired .22lr will fire if you simply rotate the cartridge and fire it again. That's because the primer compound is deposited inside the rim of the cartridge case and isn't always distributed evenly. If the firing pin hits a void, then the round will not fire, but will if you rotate the round and try again. So if that happens, explain that to her, rotate the round, and try it again.
Above all, focus on three things: 1) safety, 2) patience, 3) be a calm, steady, quiet presence.
M1911