Who or what is being redeemed in The Shawshank Redemption?

You don't have to be a hardened criminal to be in need of redemption. Andy enters Shawshank a guilt-ridden mess, unable to cope with the past. And as he says early in the film,

"She was beautiful. God I loved her. I just didn't know how to show it, that's all. I killed her, Red. I didn't pull the trigger, but I pushed her away. And that's why she died, because of me."

It takes him going through a lot of metaphorical and literal shit to come to terms with what he did and did not do regarding his dead wife, and then he produces his own penances: the creation of the library, the opera scene, inspiring Red to hope again. Thus redeemed (in his own eyes), he then begins the plot to screw over the warden and earn his freedom.

There is a theme of Redemption vs Rehabilitation best voiced by Red as a character.

Rehabilitation as Red points out is a fancy made up word to make the suits believe they are doing society a service in locking away the criminals while they ponder on their criminal acts as a punishment for their crimes.

Redemption comes in the good that you do for others. Andy reveals to Red that hope is the only thing that keeps him going. It is the hope that he can bring to light the crimes that were committed by people in charge of prisoners. Through the glimmer of hope that Andy brings to the prisoners with the small measures of freedom in playing music over the PA system. The hope that once he leaves, he can go to Zihuatanejo and live off a fishing boat. The hope that he can continue friendship with Red after they have done their time.

Indeed, there is the contrast between Red and the old man Brooks who commits suicide after leaving the prison because he had little hope on the outside. Without Andy, Red might have ended up the same as Brooks, friendless and without hope.

Red is the one who is redeemed.

Red has adjusted himself to life inside the institution and has given up hope. The story appears to be about Andy and his time in prison and his escape, until the denouement makes it clear that the story has actually been about Red learning from Andy how to hope again.

The difference between Brooks' experience upon parole and Red's is that Red has learned, from Andy, how to still have hope that he can have something to look forward to outside of prison.

It could be many things -

As you mentioned - Red, as a hardened, lifer criminal who had nothing to look forward to if we has released, and probably would have gone the route of Brooks, instead going off to finish his days in a hopeful, productive, human way - something he never had, it seems, even BEFORE committing his crimes.

Yes, Andy, as was never guilty, but was able to overcome the system that incarcerated him, and would have kept him for life, and taken his life, and also overcame the corruption of the system.

The Shawshank prison, itself, as Andy exposed the rot/cancer of corruption there, and the culprits were hauled away and replaced, and, probably, to a certain extent, the prison itself was reformed. Not sure why you'd think it would get worse with the head of the guards arrested and the warden killing himself, and the corruption and abuse exposed in all of the public newspapers. And that doesn't include the prison library that Andy built out when he was still there.

Take your pick.

FYI - the original novella upon which the movie was based was Steven King's Rita Hayworth and the Shawshank Redemption. I was unable to locate any articles where Mr King, himself, addresses who or what is redeemed.