If I give my sister a gift of $14,000, do I have to wait until after April 15, 2016, to make another tax-free gift?

No. You can give each person up to the annual exclusion amount ($14,000 in 2015 and 2016) without any gift tax, or even the need to file a return. A person can make one gift on December 31, 2015, and another on January 1, 2016. However, there is no carryover of an unused exclusion amount. Thus, not making gift in 2015 does not increase the gift limit for 2016.