Not as dramatic as the forest park or the mountains, but that's the point. The Shimna River walk through Newcastle town is the gentle option: trees overhead, water beside you, nothing required of you. The path follows the river from the town centre through a stretch of mixed woodland before it opens out towards the shore. It takes about 45 minutes at a slow pace.
Good for families, good for a reset between bigger walks, and free. The Shimna runs all the way through Tollymore Forest Park further inland, so if you've come from the forest park this path is a quiet continuation. It's also useful as a warm-up before tackling the Mourne Mountains from Newcastle, or as a decompression walk afterwards. Not spectacular, but pleasant in the way that easy, well-maintained green space in a small town tends to be.