Cats can be very hard on door and window screens -- they like to sharpen their claws on the screen material, and in doing so they tear up the fabric or wire mesh of the screen. It's essentially impossible to train a cat to stop scratching screens, particularly if more than one screen is involved. The best way to deal with the problem is to block off the screens so that the cat can't have access to them. There are several ways to approach a screen covering project without buying commercial screen guards.
One way to pet-proof screens is to replace old screens with sturdy, pet-proof screening. Most people should be able to do this, though it will probably require changing to a smaller spline, since pet-proof screens are thicker and heavier than conventional screening. The fiberglass screening material is available in different widths to fit almost any door or window. Experts recommend that you roll the new screen into the spline groove before you insert the spline.
Use clear acrylic panels to cover the lower part of a screen door or a window screen to keep cats from being able to claw it. You can usually purchase these by the foot and cut them to desired size at the store, though you can cut acrylic at home without too much difficulty. Attach the plastic section directly to the screen's frame on the side that needs protection. Acrylic will block the airflow where it covers the screen, but it doesn't block the light.
A metal grill that fits over the screen works well to keep cats away from a window or door screen. You'll find ready-made units on the retail market, but a homemade version is simple to construct. The main requirements are a piece of metal grill fabric the right height and width, and screws to hold it in place. Typically you can purchase this material at a building supply center and have it cut to order at the time of purchase.
For basic, immediate protection, you can cut a piece of wood to fit the screen so a cat can't access the screen material. This is most practical as an immediate, short-term solution such as when a cat's safety is at stake. Affix the wood directly to the screen's frame or to the window frame around the screen. A piece of 1/4-inch plywood will be adequate; it's ideal for blocking off cats from high windows and compromised screens that may allow them to escape.