I have no kids, but I'll post about things my Mom taught me as a very young child.
I used a piggy bank (with no stopper plug) for "found coins" or earned coins (lemonade stand, nightcrawler sales to a neighbor) instead of spending it. I could take coins out anytime I wanted. I made the piggy bank in ceramics class. I still have it.
When the piggy bank got full, then I would roll the bank back and forth to get all of the coins out. I counted it myself. Whatever was in the full piggy bank went into a savings account.
Any babysitting, cleaning, sidewalk snow shoveling, 4H county fair prizes, etc was deposited for me by my Mom. She would show me the passbook so I could see the balance. I was allowed to keep a little of it but the majority got deposited.
If I needed money from the savings account, I had to come up with a better reason than candy or soda. If the reason was good, Mom would take me to the bank to do the withdrawal on her next day off. If the money was needed immediately, she would do a "loan" from her pocketbook and do the bank withdrawal on her next trip.
Once I was a teenager, I kept track of my earned pocket money myself. I still gave money to my Mom for my savings account. I continued to do so until I was 18 and could go to the bank by myself.
I never had too much saved, but I learned the habits of saving, accumulating, and depositing while I was still in grade school.