Tollymore was used as the filming location for early Game of Thrones sequences, the Haunted Forest, the first wildling encounter. Even without that context, it's remarkable: old-growth trees including ancient beeches, Gothic stone arches built in the 18th century by the Roden estate, and the Shimna River threading through it all with a series of bridges crossing it at different heights. The forest has a quality of light that changes significantly with the weather, and in mist or low cloud it becomes something close to extraordinary.
The lower river trail takes about an hour at a comfortable pace. The climb up through the upper forest into the foothills of the Mournes extends it to a half-day. There's a small car park charge. Go on a weekday if you can, weekends in summer can bring more visitors than the trails comfortably absorb. The arches along the lower path are the most photographed feature; the upper forest is quieter and rewarding.