Hello, everyone!
This is my first time playing the KAO series of games.
So, I was wondering if you could save your progress in Kao the Kangaroo 1?
I know there are checkpoints but I realize there are only for levels and not for the overall game.

You can save the game by pressing ESC and then choose save game after you beat a level at the level selection screen.
Also checkpoints are usable in levels where you drop them with enter.
It's useful to save and then reload if you lost too many lives, since game over by losing all lives just brings you back to the starting screen and if you didn't save then tough luck from starting from the beginning.