I got a social security number for them (within 30 days after they were born), then I set up a regular self-directed brokerage account with myself as the legal guardian (I think it was under the "Uniform Gift to Minors Act" rules) which allows me to direct the investments even though the account is under their social security number and is taxable to the child.
I then bought the stock(s) I thought they would be interested in and would be familiar with what the company did (Disney is a great example) and reinvested the dividends.
I'd add to the account each year following the IRS annual gift exclusion rules and then would buy a different stock, such as Coca-Cola, and reinvest the dividends.
By the time they started college they had over $50,000 in their brokerage accounts. And more importantly they could intelligently discuss PE ratios and dividend yield.
This post was edited on 3/13 at 9:40 am