Transferring directly from one phone to the other, without using anything else, doesn't really work so well. Your best options are to either ...
1. Copy the images over to the SD card and then move the card over to the other device. (doesn't always work)
2. Upload all the images to a cloud or google account, then re-download them onto the other device. Make SURE the photos are on the new device before you delete them off the old one as sometimes there are issues in the moving (to cloud accounts or google accounts.)
3. The simplest, and least problematic way. Plug your current phone into a computer, via USB cord, and copy all the images to the computer. Then connect the new phone to the computer, via USB cord, and copy all the images to the new phone. Make sure they are all on the new phone, and then delete them off the old phone. I would also suggest leaving a copy of them on the computer, that way you have them, should your phone break down, get lost, stolen, etc.