How do I protect the screen on my screen door?

If you are trying to protect it from a pet, they sell screen door guards that stop the pet from running into the screen or denting it.

I used a piece of plexiglass over the bottom area of my screen

Any home improvement center or hardware store sells screen guards.

There is a product called hardware cloth. It is basically strong wire mesh and comes in a couple of sizes of holes. It's easy to cut with wire cutters. Attache it to the door frame over the screen where you need to protect it. Kitties and doggies might bend the hardware cloth, but won't rip the screen. A very determined child could rip the screen by poking things through the holes, but it would not likely happen accidentally.

Hardware cloth is a pretty useful product to have around in general, too. Attached to the back of a picture frame, it makes a great place to hang earrings. Bent up into box shapes, it makes a nice basket. And I'm sure there are many other great uses.