This app was mentioned in 11 comments, with an average of 1.18 upvotes
I've never used the Samsung Account backup, but I have used MyBackup Pro to backup my SMS and MMS messages off of previous phones and restore them to new phones, or after doing a full wipe. It works wonderfully.
https://play.google.com/store/apps/details?id=com.rerware.android.MyBackupPro&hl=en
You can back up the apps on your current phone with an app like MyBackup Pro and restore them to the other. I've been dragging XiiaLive after me from phone to phone for many years now. :)
I've always been a fan of My Backup Pro. Used it for many years https://play.google.com/store/apps/details?id=com.rerware.android.MyBackupPro
There are so many ways to do it through Apple, or Google, other online backups. Including my personal favorite MyBackup, to computer, backed up to a NAS, and the NAS backed up online.
https://play.google.com/store/apps/details?id=com.rerware.android.MyBackupPro
The term is 'migrate'.
The most complete backup/migration app for non-rooted phones is MyBackup Pro (pay version). Install it on your old phone and back up all categories to your computer (or you can buy cloud storage space). When you get your new phone, install MyBackup Pro from your Google Play account and restore all categories. Follow the user guide. There are just a few files that MyBackup Pro won't backup, but for the most part you get the things as they were on your old phone.
If your phones are rooted, you can copy all files, to get a complete migration.
Your phone store can migrate your contacts and some data, but will likely miss some of the stuff the app will handle.
https://play.google.com/store/apps/details?id=com.rerware.android.MyBackupPro
Best money you will ever spend.
There is a feature that if both phones are in the same WiFi network, it will zip the backup on one phone and automatically transfer to the other one and restore.
Backs up just about everything.
For contact information you should be using your Google account to have it cloud stored. If you do, all you need to do is login to another device with that gmail account and it will automatically restore the contacts.
For photos, use this.
https://play.google.com/store/apps/details?id=com.google.android.apps.photos
Select backup at highest quality. This will auto sync with Google and all photos will be backed up. Great app.
Use MyBackup Pro to easily do this. You can choose to just only backup the MMS images, it will then zip them up and place them on local (and online storage if selected) afterwards. Then when you unzip the file, all the MMS images will be in there.
Another great app that saves both SMS and MMS (free) is SMS Backup & Restore, however I don't think it saves the images separately (each its own file) like the above solution does.