You can gift up to $14,000 worth of stock to a child with no gift or estate tax issues (The gifts does need to be a gift of a "present interest").
Your spouse could gift the same amount to a child.
If you gift more than the limit, then you need to file a gift tax return. You may or may not owe gift taxes. But, unless we're talking about a gift in the millions of dollars you don't need to worry about that.
The gift is valued at the Fair Market Value on the date of transfer. The basis (the value you use to calculate gains or losses) is the same for your child as it was for you. The holding period of the stock includes the time you held the stock.
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