Another option is Dropbox.
Dropbox (see my signature to sign up) will do the same a G+. Once you set up your Dropbox account and enable "Camera Upload" on your old phone, wait for Dropbox to finish syncing all your photos (need to be on WiFi unless you change settings to sync on mobile data). Then when you start using your new phone, install Dropbox and your photos are in the Camera Uploads folder of Dropbox. You can even click the arrow next to the folder to Move the photos to your internal or SD card storage.
Make sure you turn Camera Upload on in Dropbox on your new phone to backup your photos any time you are connected to WiFi. Also, as with G+, you can access your Dropbox folders from any browser, anywhere, or via the Dropbox app on your desktop, laptop, or tablet.