North Carolina is absolutely chock-full of gorgeous natural parks and trails. Do yourself a favor and check out Chimney Rock State Park and Crowders Mountain State Park. Both offer hikes from difficult to easy and tons of landscape changes so you'll never get bored.
If you and your dog are the active types, seek out Turbo Paws Flyball in Charlotte. It's a new drag race-type sport with lots of hurdles and running that goes along with fetching and placing balls. It's high energy, super fun, and if your dog takes well to the classes, they also have a competitive team your dog can try out for.
During the summer months, Summerfield Farm hosts Music Mondays, where you can grab dinner and listen to some great local music, and Terrace Thursdays, where you can catch great views from atop a refurbished barn. There's lots of outdoor seating for you and your pup, cold drinks, good grub, and some space to run nearby if need be!