The gift tax exclusion is per person on both the giver and the recipient side. So you and your spouse could each give $17,000 to one person (such as your child). You could also give an additional $17,000 each to your child's spouse, if they are married.
As mentioned, gift tax is never actually owed unless your lifetime gifts and estate are more than $13 million. The exclusion just determines whether or not you need to file a form. If you want to give a larger gift, you just need to fill out form 709 and mail it to the IRS, you won't actually owe tax.
