Welcome to the Android Forums. Sorry for the delay in responding to your question.
Photos should be stored in a folder called DCIM. Transfer the contents of that folder to your computer. Then transfer back to the S4 (If no DCIM folder yet, take a picture with the camera).
If you want to avoid connecting to a PC, you could use a service like Dropbox or Box and using their app, transfer the photos to the cloud, then install the app on the new phone and transfer them back.
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