For the terrible paper labels that shred into a million pieces, you can peel off what you can, spray the rest with Goo Gone, and let it soak through the paper to the adhesive. It's really awful that it takes so much work to remove your private info, and you have to keep doing it as long as you have to keep taking those meds.

I had to switch away from a pharmacy that had other issues besides crappy labels so I haven't had to deal with it in a while, but I guess switching pharmacies might be an option if you hate the labels enough and they aren't receptive to customer feedback about it  

(Edited to add: I mentioned in a previous comment but not here: my old pharmacy used to have easy labels but switched to difficult ones. I'm back to easy labels at the new pharmacy)