Legend of Zelda: Skyward Sword HD has been released on Switch, and for those of you who haven't played it before, it's a different kettle of fish. Saving is very reminiscent of Majoras Mask for the Nintendo 64.
To save in Skyward Sword HD, you need to find yourself a bird statue. This is the only way to save manually. The bird statues are quite hard to miss, as they are quite tall and some parts of them glow with green light. The first one you see is right across from Knight Academy.
All you have to do is interact with it by pressing A and then selecting the save button. Choose the save location you want to save to, then select Save again and the jobs are done.
Related: How to switch to button controls in Legend of Zelda: Skyward Sword HD?
If you don't want to play anymore and want to save and get rid of it. Then select Save and Exit, then follow the instructions above, and after saving it, you will be returned to the game intro screen, and all progress will be saved.
There is an autosave in Skyward Sword in case your game crashes or you forget to save. Automatic backups are usually pretty good at backing up, but they will still only back up at selected intervals. It will not automatically record continuously.