Read the release notes for the new server version to which you are migrating, including all of the intervening releases. Pay attention to the incompatibilities section carefully; PostgreSQL does change from release to release. Assume this will take some hours.
Set up a test server with the old software release on it. Restore one of your backups on it. Upgrade that system to the new release to verify that there are no conflicts from software dependencies. Test your application. Make sure you identify and test each add-in PostgreSQL module you were using to confirm that it still works at the new release level.
Back up your production server. Prepare for the worst; hope for the best!
Most importantly, work out who you will call ...
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