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how do I save a game?

What do you mean? Games are saved forever in your account and you can look them up at any time. There are also features to filter and find them.
If you want, you can star them (the star is hidden somewhere in the upper left corner), so you're also gonna have a bunch of your starred and preferred games.
If you want, you can import the game in a study (analyze game>left menu>create study) that you can then edit, make public or private, and so on.
You can have the pgn of your game and analyze your game locally with an offline chess program.
If it's not enough, you can download all your games locally and analyze the whole database.
If you mean exporting a game or multiple games @Pickler

For exporting the PGN of a single game, ypu will not find the link untill you have hit the Analysis button, so it is somewhat hidden at first sight (IMO).

To download all of your games PGNs at once click the little download button on your profile page above your profile info top right corner.

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