If it came down to it the woman needs to be able to shoot it so a lesser kick would be ideal but mostly im looking for cheap.
A (rather mad) retired sergeant major thought it was great fun to start me shooting at a very early age. "Only thing a woman can best a man in, kid". I came very close to fainting from pain once or twice, but I had put the stock to my shoulder wrongly. Prone position, steel shod military butt, no rubber pad. I've never been much bothered by recoil, otherwise. He started me off on full size military rifle caliber. I shot my first 300m rifle competition when I was 13, or maybe I had just turned 14. I've shot all kinds of things through the years, and also competed with pistols. But I've remained partial to bolt action rifles and long ranges. Most Remington 700 series are very accurate and good value.
All that said, considering both ammunition costs and range requirements, I think you should get yourself a .22LR, as a first rifle. I also think you should get yourself a bolt action. They are easy to understand, and inherently safe (unless you're stupid enough to keep a round in the chamber).
The most important aspect of shooting is familiarity and practice. .22LR will translate into a lot of practice because it's much cheaper than centerfire cartridges, and you don't need long ranges.
The most important aspects of practicing, are safety routines and trigger pull. Learn how to do a proper trigger pull and practice it always. It takes years to develop a really expert trigger pull, but some 100 rounds a week, will get you basic competence in just a few months (provided you consciously practice it). Idiots jerk the trigger when they feel the sights look good. They never learn to shoot and the world seem to be full of them. They stare at you as if you come from Mars, when you put all your shots into 1½ inch group, offhand - .38 revolver - 25yards, and have no clue to how it is possible. (- It's all in the trigger pull, morons!)
Please, do yourself a favor and learn to do it the right way, from the start! People seem to have enormous difficulties, getting rid of bad habits later.
So I agree with yellowfiero - .22 bolt action.
I don't know much about cheap .22 boltactions. Kimbers are sweet, but kinda Rolls Royce market segment. Ruger 77/22 if you feel like spending, I guess, but I would vaguely point you in the direction of Marlin and Savage. But I really don't know anything about them.
Auto .22 are of course cheaper though. Ruger 10/22 is the obvious choice, as has already been recommended here a lot. But I don't like autos. You have to pay a lot of attention to handle them safely.
If you want a rifle for out doors protection, rather than sport/competition/hunting, - lever action .30-30, or .44 magnum (recoil is about the same, but .30-30 is more powerful). They are less rugged, cruder, less accurate, but more speedy, than bolt actions.
For home defense, make that lever action some pistol caliber, .38 or .44 special. .44 Magnum is way too much from a rifle barrel, indoors.
There's no point in any military assault rifle. Triggers and sights are so bad, you'll never learn to shoot properly. They are also ungainly, crude, ugly, inaccurate and socially offensive.