2013 prius, passenger fender liner replacement, what parts do I need?

Find yourself a parts site with decent pictures, start pulling the old one out, note all the fasteners as you go, find them on the site. Or go to a dealership parts department, they should be able to help.

There's one tricky plastic fastener on the edge of the wheel well, around 9-oclock as you face the fender. It's got a slot on it's head: you turn it 90 degrees, then it's possible to coax it off.

At the front of the fender liner there's a 3 bolt connection to a plastic piece a the underside corner of the bumper. You'll likely need to replace that as well: in my experience those bolts were hopelessly rusted junk, might as well be welded.

I've had a fender liner (with that connected front piece) almost off the car: there's one screw-in fastener at the very back, first a plastic shoulder piece with a wide flange pushes in to a hole in the bodywork (capturing the fender liner), and then a screw goes in. You can remove the screw, but this shoulder piece you pretty much need to destroy to get out.

In this pic the roughly foot square piece touching the floor is the bit that's "connected" by 3 bolts, but they are hopelessly rusted crap, you might as well treat it as one piece. Just this one connection comprises 3 bolts, a strap spanning all 3, and several small fasteners that hold the strap to the plastic. Also, the whole thing's still attached to the car by that one fastener at the very back, that you pretty much have to destroy to get out:

 

If you're comparing left and right side, there are a few plastic rivets that are permanent, basically construction rivets. They wouldn't need to be ordered, are just lumped in with the large piece. They're pretty obvious, are un-removable.