I had a Samsung Galaxy SIII about 5 years ago, and back then, there was a software called Samsung Kies that allowed you to transfer content to your computer for storage, or to later transfer to another phone. I assume that's still around. It used to be preloaded into the Samsung phones, so then you just had to download the desktop version to transfer things from your phone to your computer. You can also look on the app store for "photo transfer app" or use the Google Backup which automatically backs up your photos onto your Gmail's Google Drive (I think this can be engaged by going into your phone settings/backup option and ticking one of the options that allows every photo you take to be backed up). If not, call your carrier for troubleshooting or go to one of your carrier's stores so they can assist you. I have also had luck just plugging in a USB cable to my phone and computer, browsing the phone's content, finding the folder with the photos, and dragging it/copying it to my desktop.
ETA: Forgot to mention you can also call Samsung for troubleshooting. I have a Sony phone at this time and when I had software compatibility issues with my computer, I called them and they did troubleshooting with me.