If you are on Android 4.0 or up, you can use the
adb backup resp.
adb restore command even without rooting your device. The syntax is:
adb backup [-f <file>] [-apk|-noapk] [-shared|-noshared] [-all] [-system|nosystem] [<packages...>]
To do a full backup, this would e.g. mean
adb backup –apk –shared –all –f /backup/mybackup.ab (resp.
adb restore /backup/mybackup.ab to restore the backup on the new device). But keep in mind this would mean "all-or-nothing" -- you cannot restore single apps this way. So you rather might want to include the package names of the apps to transfer. To find those, e.g. look up the apps on Google Play -- you'll find the package names in the URL (
id=<package_name>). Your backup command then would look like:
adb backup –apk –shared –f /backup/mybackup.ab com.foobar.app1 org.foobar.app2 net.foobar.app3
Restore stays the same, as
adb restore always restores everything from the backup file (no selection of parts available here).
For a list of available ADB commands, see e.g. here.