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If you have a Smart TV, then the icon for Prime app may already be pre-installed along with others along the bottom or side of you sceen depending on how you have it see up. If not, look in the apps menu and add it to your list.
First time you use Prime you will be asked to authorise it which will also synchronise it to your computer or other device, so have that handy. It tends to be easier to do this on a device keyboard. TV input screens are apt to be laborious having to tab from one letter and number to another.
That's all you have to do. Then you can watch on whichever device you chose. You can even switch between them for the same programme and the app will track where to resume.