How to restore deleted files from an internal HDD which has just undergone a full disk format?
Now my question is how can I recover all my files (properly) without
  the help of any recovery software!


You can't.  Unless you have the skill to read/edit the partition tables as well as search each and every sector for orphaned blocks which you will then rebuild, you must use recovery software.

Additionally, that's predicated on if it's a traditional (spinning) hard disk.  If it's an SSD, the data is gone.

You will also want to review this answer on backups