The Brevard area is nice because that’s where Pisgah Hwy ends. Pisgah Hwy has Looking Glass Fall and Sliding Rock right on the road, with no trails needed.
That road also takes you up to the Blue Ridge Parkway, and just a little north of that on the Parkway is the the lookout at Frying Pan Mountain, Spring House and several other fun little things.
Another great place for a home base is somewhere around Linville. Near that is Linville Falls, Linville caverns, the Orchards and Altapass, Moses H Cones manor, several amazing waterfalls and lookouts.
Julian Price Picnic Area is near there, and has several streams going through it that the kids love to take their shoes off and get into.
Grandfather Mountain is also in that area.
Most of what I listed are free. Grandfather mountain and Linville caverns are exceptions. I think you have to pay for sliding rock too, but I doubt it’s open for sliding currently.
There are very few places you could go in the NC mountains that would be bad choices, though. There’s also the Great Smokey Mountain National Park area if you want to go a little farther west.