So I have been trying to get my shower drain cover off so that I can remove all the hair but I can not seem to figure out how it is supposed to come off. It doesn't have screws, doesn't twist, and doesn't pry off. I am out of ideas. Any tips would be appreciated.
Having just purchased a brand new drain to install into a brand new shower pan, I would suggest that this should, most likely, unscrew.
However, since yours isn't brand new, it likely is rather stuck and needs some help. It's apparent that you've made some attempts at releasing this and have managed to bend it a bit so further care is necessary in attempting to remove it. You'll want the largest flat-blade screwdriver you can find that will fit into the slot right at the center and give it a twist (lefty-loosey) to unscrew it. If that still won't budge it (or feels like you're going to start damaging the strainer) move on to the next steps...
I'd suggest some small amount of drain cleaner carefully poured between the strainer and the trim ring and allowed to sit and soak for a while, then rinse thoroughly (you don't want to splash that on yourself while you're struggling to get this off). That should help break up whatever gunk is binding the threads. You may need a couple of applications. Note that whatever manages to spill down the drain will help break up whatever is clogging your drain. ;)
If a couple of applications of that don't help, you may need a penetrating fluid to work its way into the threads. Many people use WD-40™ for this, though I've found that PB Blaster™ to be more effective on crud-encased, rusted bolts on vehicles. Whatever product you choose (there are more), carefully read the instructions to ensure that it won't damage your tub/shower floor. If it's porcelain over steel, it should be impervious to just about everything, but if it's fiberglass, acrylic, stone or other material, you don't want to put something on there that will stain it or eat away at it.
Once you've got the strainer out, you may be able to hammer the little dimples flat if you have a hammer and an anvil (or something that will work as an anvil). You may want to protect the visible surface finish by using a rubber mallet or a piece of wood if you only have a metal hammer.
This drain seems to be mostly used in mobile homes and RV's. The strainer when new is simply is pushed into place. Until it no longer stays in place because the person that last removed it deforms it (the strainer). Some plumbing forums have some unkind words for this person because of what they do next. Instead if gently reforming the strainer so that it fits the opening the way it should (as mentioned by freeman). They put silicone around the opening where the strainer sits in the drain and press the strainer back in. This makes it nearly impossible for someone to remove without damaging it. I will try scrapping the silicon out with the tines of plastic forks or a X-Acto knife