Picasa has been retired, I've used an old version until recently but cannot recommend it. Google photos is an online site that invites you to upload everything so that they can do what they want with it (my view...).
Your question made me rethink my own image sorting needs, and I've gone ahead and purchased the latest version of ACDSee for my Macbook (I've got ACDSee Photo Studio 6). I very quickly fell in love with it again.
- Main interface offers folder tree on the left. 'Tiles' with thumbnails of images is the main view and there are additional options to show along the right side of the screen.
- You can select one or more images and right click to 'move to folder' or 'copy to folder' as needed.
- There is a function that searches your drive, or a subset of folders, for duplicates of an image (can take a while).
- I've used batch rename on many occasions. There are other batch functions included in the software.
- The software also includes various editing options, but browsing and sorting through the many images I've got stored on my harddrive is the main thing I use this for.