Netflix has an excellent help section, but I'll share some screenshots (from my iPhone) to make it easier: https://help.netflix.com/en/node/54816
First, make sure you have the latest software update that makes downloading possible (see photo above): App store, & click "Updates" on bottom right.
|Log in, then click at the top left...|
|Not all shows can be saved offline--try "Available for Download"|
|When you find something, it will look like this. Tap on the arrow at the right to download|
|Saved offline downloads look like this|
|To play or delete, just tap the blue arrow|
Check out the blog, cheapskates-on-move-tv-movies for info on mirroring your iPhone or iPad to the TV.