A few things to research are the Hiking trails at the falls themselves, go to all 3 lookouts, they're all worth it! If you pack a picnic and spend some time sitting and soaking the view in, then it is a full day activity. It was for us. There is another trail that goes down to the base of the falls. It's fairly strenuous, but was an adventure for us and a stunning spot to hang out.
Go to the Linville Caverns....on second thought, if you're in Kentucky, you likely have seen better caves, we haven't.
There is a drive to an overlook that follows a fairly rough gravel road for about 6-8 miles. It was the most stunning spot we've seen in all of Appalachia. I'll find out the name for you if you message me before going. Or perhaps someone else knows?
The Linville Viaduct is worth seeing, but was actually a let down for me. It is an engineering marvel if that's your thing.
And of course, Grandfather Mountain itself! They have a good website that gives an idea of all it offers.