As parents, we're the first and most constant teachers our kids will have. That can be a daunting responsibility sometimes, since we may not feel entirely prepared to help them with some of the lessons they need to learn, but that doesn't mean we get out of doing it. For example, an understanding of money and basic financial ideas is something that kids can, and should, start developing within the family. Financial literacy is necessary in order to learn and develop financial responsibility.
Kids get interested in money at a young age, from what I've noticed, and a parent can use that interest to teach basic concepts, including math. My son was a "little capitalist" in elementary school, and my stepson is the same way -- if the interest is there, a parent can run with it.
Relatives and friends often give money gifts to children; you can introduce them to the bank by opening a savings account for them and taking them there to deposit birthday checks and the like. At the same time, though, kids need to learn about money as an everyday thing. Provide kids with a container to use as a bank for cash at home; I have yet to meet the child that didn't love to put quarters, and even dollar bills, into his own money jar and watch it fill up.
But saving doesn't just mean putting money away and keeping it somewhere; you can take opportunities to teach saving with the intent to spend on specific goals, too, such as a special outing or a computer game. The practice of saving up for something you want can't be reinforced enough these days, it seems to me.
Taking kids on shopping trips helps them learn how to determine whether something's worth spending their money on -- encourage them to bring their own money with them! -- and reinforces the saving-up lessons. Just because they have the money and can buy something doesn't mean they should.
Savings contributes to net worth, but day-to-day money management is based on income and expenses. An allowance can be a very important tool in teaching finances. Some parents tie it to household chores, some don't -- that's a family-policy decision rather than a financial one. However you do it, it helps them learn about earning money for themselves.
Older kids can use some of their earned money to start paying for some of their own expenses, such as lunches and entertainment, while parents continue to provide the essentials. There are also more earning opportunities for older kids; they can do more complex household jobs, provide services to neighbors such as babysitting and yard work, and eventually get part-time jobs. At that stage, it's a good thing to help them learn about what's involved in making a living and paying one's own way, and parents can get creative with this. One mom went through the newspaper with her teenager and picked a few jobs; they looked online for typical pay rates for the jobs, and then figured out how to live on that money in their area, considering costs such as apartment, transportation, and groceries. Another parent teaches the strategy of thinking of the purchase price of something in terms of how long it takes to earn the money to buy it, which reinforces consideration of the value of time as well as money.