How to save a users progress in quiz game

The usual way, assuming there's not a huge amount of information to save, is to store it in user defaults:

https://developer.apple.com/documentation/foundation/userdefaults

You can save all of the information in a blob under one key, or split it up into values stored in multiple keys, as you wish.

iOS? The game might be played by the same user across different devices, so, take a look at using

NSUbiquitousKeyValueStore

to store the game state in the user's iCloud data. That way, game status can continue where left off regardless of which device they use, and protected should they delete the game/app, which won't occur using defaults.