My favorites are:
Pisgah: Do Looking glass falls, sliding rock, and Looking Glass Rock Hike
Dupont State forest waterfalls Dupont State Forest
Craggy Gardens (beautiful drive there on the parkway) Craggy Gardens
Greybeard & Seven Sisters (in Montreat) Greybeard
Mount Mitchell Mount Mitchell Trail
Black Balsam Area Black Balsam Trails
Chimney Rock/Lake Lure area: I love to take a day and do all three of these, Bearwallow Mountain Hike on the way there (from Asheville), onto Chimney Rock State Park, and finally a swim in Lake Lure.
If you want a quick hike, you can stop anywhere along the Blue Ridge Parkway and hop on the Mountains to Sea trail (marked by a white dot); Nice winter views and easy access from AVL.
The picture below is the Snowball Mountain trail which is just off of the Mountains-to-Sea trail in the Craggy Gardens Picnic area. Go to what looks like a cul de sac and park. Trail is to left – only 2 miles to lookout but very challenging!