Traveling to Great Smoky NP for the first time, any good recommendations for hiking?
Hello! There are too many great trails to hike at the Smokys. I was there with my family back in March for 8 days, and we didn't even do half of what we wanted to do. My suggestion first of all is, if you like longer hikes (5-10 mi), read about Mount Le Conte and see if you'd like to go there; start with this since is difficult to get a reservation for lodging or using the shelter overnight, because of its popularity.
That out of the way, my other suggestion is, if you only expect to do day hikes, drive around and across the park. Going across the park (e.g. Gatlingburg to Cherokee) is a nice drive with many potential stops along the way and access to great trails. The Roaring Fork Nature Motor Trail close to Gatlinburg is a beautiful drive with several trails not terribly long (my family and I did the Grotto trail, about 5 mi round trip) and historical farming buildings along the road. By all means go to Clingman's Dome, to have a 360-degrees view of the whole area (I recommend going there as early as possible, very popular).
Another nice drive with lots of hiking opportunities is Cades Cove, west of Gatlinburg, a nice valley surrounded by mountains. The park entrance close to Cherokee south of the Smokys has some historical farms as well, and if you drive west to Deep Creek, you'll find many trails that are more remote and secluded. I backpacked the Deep Creek trail up the mountains for a couple of days with my family and found very few people (four total). At the campsite we stayed, #56, we were all alone for two days. That was kind of a luxury for us.