I like my CenterPoint AR22 Series scope. I have it mounted on my 10/22. I paid around $55 for it at WalMart.
It's 3-9 x 32mm with Red/Green illuminated Mil-Dot reticle. It comes with 3/8" dovetail rings, but if the Remington uses some other mounting, Walmart has a pretty decent selection of rings. It also comes with flip-up lens covers.
These 597's really are good little guns, especially for the price. I know what you mean about breath control, etc., though... mine has the polymer stock and regular ("noodle") barrel, and it is so light that it can be hard to keep steady, especially at 100 yards. I have a bi-pod on the front of mine, now, and that helps a bunch, and now I just need a sand-bag for the buttstock for those long range shots. Still, not too bad when you are making shotshells dance at 300 feet. Squirrels beware!