How to get pictures from old iPhone to new iPhone?

I just got a new iPhone and I need to put the pictures from my old iPhone to my new one. I had to make a new apple ID/ iCloud, so those pictures were not on a photo stream. I have plugged my old phone into my mac and have all the pictures imported, but since my family was on the same apple ID, I also have their pictures. I need to put only MY pictures onto my new phone. Verizon made it sound much easier than it is

The easiest way is to backup your old iPhone with iTunes/iCloud and then restore it to your new iPhone.

You may try this phone to phone transfer, it can help you transfer files between iPhones. You just need to connect your two phones to computer through USB cables, then you can drag and drop the pictures between the two phones. This software works fairly well for me.