Encourage your child to save half of any money that they receive, whether it is from their allowance or a gift. Work with them on how they would like to save the money. Would they like a piggy bank or would they like to open a bank account? Have them participate in dividing the money and making the deposit.
Talk with them about what they would like to save for and how they would like to spend the other half. If you keep them involved and make it fun, they will be more willing to save their money. Teaching them to be financial smart at a young age will pay off as they grow up.
I have 4 young kids that each have a piggy bank for collecting coins and putting their allowance in. When the banks are full we take them to the bank and deposit them in their accounts, usually around twice a year. When there is an extra chore to be done, I will give them a quarter or 50 cents to do it and that also goes in the bank. When they get their allowance it is also in coins, and I was finding that I kept having to go to the bank to get change to pay them their allowance and for the chores.
I finally figured out that when the banks are full, we count out the coins and I write each child a check for the amount they have in their piggy bank and we deposit that in their real bank account. Then I keep all the coins that they had in the piggy banks to pay them for chores and allowance for the next 6 months or so. It works great and they have gained so much from having their own money to spend at Christmas, at yard sales and when they want to save up for something. This way it's just a little easier on me to keep a good thing going.