I lived in Asheville for thirteen years before having to move to charlotte for my dh’s job. Very sad. We loved driving on the Blue Ridge Parkway which was only minutes from our home. Our kids loved the WNC nature center. It is a fun and inexpensive place to visit. It is right off i-240.
Of course, the Biltmore House is amazing and a must-see. They have a barn area with lots of chickens, goats, horses that the kids will enjoy even if they don’t get much out of the house. I’m not sure where west of Asheville you will be but there are so many places to hike and see waterfalls. Outside of Franklin, there is a gorgeous drive up to Highlands, NC which is itself a beautiful little town. The drive up is full of waterfalls. In fact, there is one that you can actually park and walk down and behind the waterfall. So neat. We love that one. Once in Highlands, there is also an actual hiking area that leads to other waterfalls.
If you go to Flat Rock, you can go to Carl Sandburgs home. Its free to just peruse the land. There is also a barn there with goats, chickens, etc. I had a friend that went there with her kiddos and they all loved it.
Enjoy your trip. You will fall in love with western North Carolina!!