The .257 is a great round with very minimal recoil, however, the .243 is hard to beat. The .257 shoots a 115gr bullet I beleive and most folks shoot an 80gr or so in the .243, so the .257 has a little more meat to the projectile. I think the .243 probably shoots flatter and would be my choice. As far as knock down, well a bullet does not actually knock down a deer. The direct shock to the central nervous system is what causes a deer to collapse, not the impact force of a bullet. The more energy transferred to the deer by the projectile betters your chances of shocking the nervous system and putting him on the ground. I am not certain without looking a ballistic table if the .257 delivers more energy than the .243.
If you are going to step up from the .24 caliber to a .25 caliber, would look at the .25-06. Would not be much difference in recoil. Either or will get the job done on deer. The .25-06 and .243 both have better velocities, lesser drop, and better energies than the .257.
Might look at this ballistic table for comparison and you will get a better picture http://www.remington.com/products/am...=9&c2=13&c3=10. Just click on the loads you are interested in and compare. My wife and daughter both shoot 100 grain rounds from .243's. The wife uses remington corelokts and my daughter shoots winchester power points.
Recoil will be affected by the load and the rifle, but here is some general info on recoil energy.
.243 - about 8lbs
.260 - about 10lbs
.257 - about 10lbs
.25-06 - about 12.5lbs
.270 - about 18lbs
As you can see, there is a considerable difference between the .270 and the others. I am a short action fan and would not reccomend the .270 for a woman. You can load the .260 with 100gr, 120gr, and 140gr bullets, so it is a pretty versatile round for coyote,pigs,pronghorn, and deer.
The roberts does have more power than the 243. But the 243 is still going to be cheaper to shoot.
Not sure for handloading, but in factory loads, the .243 has better performance in deer bullets than the .257 according to both winchester and remingtons sites John, and is considerably flatter shooting over longer distances. As mentioned there are plenty of options with the .243 too, and it is good to know you can go to pretty well any wal mart and get ammo. Remington and winchester both have just one load for the .257 which is a 117 grain core lokt or power point.
If you want to step up to a .25, look at the .25-06, but the .243 is good enough to hunt deer with.
I see that a couple of folks brought up a good point about ammo cost and availability. The .243 would be your best bet for those reasons, I recommended the .260 because of the versatility to shoot 100gr on up to 140gr. Like I stated before, it is hard to beat the .243. I also noticed some folks reccomending a couple of larger calibers ( ie: 35, 308, 270 ), while they may not think the recoil is too heavy , I don't think they are considerin that we are looking for a rifle for a beginner female. I am blessed to have the opportunity to guide hunts for several children and some new hunters each year and will tell you from experience that in most cases it is best to bring them in shooting a rifle or shotgun with very minimal recoil. I definately would not consider a rifle with a felt recoil energy of more than 10lbs.
I would choose the .243 anyday. Small recoil, ammo is easy to find, and it is very flat over long distances. In fact, my dads .243 hits dead on with no hold over out to 175 yards. Definitley gets the job done on deer.