You should be able to connect your phone to your pc via your USB cable and use your file manager to drag-and-drop files.
Hi there Cwms!
I'm so happy you want to ensure your memories are kept safe. I'm not sure of which android device you have, but they're all pretty similar.
Please follow these steps:
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There is usually two spots to look INTERNAL and possibly EXTERNAL (If an sd-card is installed). Usually your looking for a folder called DCIM and in there is where the photos you take are stored. If you have downloaded or received and saved photos you will need to check both drives for a folder called download(s) or Download(s) and see if the photos are there. You will want to make sure you have your phone in MTP mode. this allows for that to come off. There are other options as well that can help back them up IE: Verizon Cloud, Google+ just to name a few but there are many of those as well..