With the $2,000, I would consider putting half into a domestic equity fund such as SCHB. Today SCHB is priced at $31.26, so approximately 30 shares could be bought. With the remaining $1,000, I would consider an international stock fund such as SCHF, priced at $28.09. Approximately 35 shares could be bought at this price. The number of shares, of course, should all be figured out by the child - it is part of the financial literacy learning.
At this point, having invested, your child has plenty to work with on the investment front. He or she can learn about the dividend yields of the funds, how to find out when the dividends are paid, the holdings in the funds, how exchange traded funds are different from individual stocks and so forth. As time goes by, even the return on the funds can be calculated.
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Two specific payoffs come from this activity. First, when he or she is initially faced with making choices for their first 401k, they will have considerable experience under his/her belt . In fact, they will be able to assess the quality of a company 401k before they take a job. Does it have high expense funds and poor investment choices? They will be able to answer this question.
A second payoff is that the fund has a lot of time to grow. If this is started at the time the child is 16, for example, from proceeds of a summer job and grows at a compound rate of 8%/year, then at age 65 just the initial $2,000 will have grown to $86,000! Even with inflation, this will be real money and, to boot, available tax free, if you believe (huge assumption) the government won't change the law on Roth IRAs!
Disclosure: This information is for educational purposes. Individuals should do their own research or consult an advisor before investing. I own the funds mentioned.