Visitors to the Great Smoky Mountains national park are allowed to pick fruits and berries for their own personal consumption. You are not allowed to pick plants, or harvest Smoky Mountains delicacies such as ramps or Ginseng as picking these plants is not only illegal it reduces its already precarious population within the park.
Blackberries should be picked when all dark and slightly soft for the best flavor. Blackberries have varying degrees of thorns so be careful picking berries and please only take what you will immediately consume and don't damage or cut the plant.
Since the berries are ripening, be careful that bears who also love these summer treats are not close by and that if you are walking into tall grass or weeds you don't steep on either a copperhead or rattlesnake which can be hiding in the undergrowth.
If you did go into the brush make sure that you check yourself for ticks as ticks are in season in the Smoky Mountains.