Dupont Recreational Forest has amazing hiking! HIghly recommend.
We've gone camping and hiking in Pisgah National Forest. Its a great place with many different levels of trails. Super close by is Oskar Blues Brewery and Sideways Farms Brewery as well.
The South Slope is one of our favorite areas in Asheville. There are plenty of shops, restaurants, and breweries to explore. We highly recommend: Bhramari, Burial, Green Man, Twin Leaf, Catawba, and One World.
Sierra Nevada is an amazing structure slightly southwest of Asheville. They have a great restaurant, which you need reservations to get into! They also have a great outdoor space and patio where you can grab a beer and potentially catch some live music. They have outdoor games as well.
The Biltmore Estates is a great place for families to explore. This historical property spans across hundreds of acres of land in south Asheville with beautiful gardens, shopping, and tasting rooms. They have onsite wine tastings, restaurants, and other activities that you can reserve online. You can also just drive out there and walk around. We highly recommend going here if you haven't been to Asheville before!