My DVR is being replaced because it quit working. How do I get the programs recorded from the old DVR to the new DVR?
Unfortunately you can't retrieve them. DVRs aren't really meant for long term storage. If they break or want to upgrade or switch providers you will lose the recordings.
You can save your series recording schedule and other settings "to the cloud." I don't remember exactly where it is, but poke around the menus and you should be able to find it.
Be sure to do it on the old DVR before installing the new one. It should give you an option to restore settings from the cloud during setup.