Western Washington has a lot lowland trails that can be hiked year round. It is easy forget that we basically live in a rain forest, until you start walking in a forest after a rainy week.
The Four Ceders Tail is located inside Tolmie State Park. It’s a two mile long trail that winds through the park. It’s all forested, and it is has minimal elevation gain. This can be hiked with kids and dogs (leashed)…and I guess even goats. The day I went hiking, there was a group of people walking with their goats! It was pretty muddy on parts of trail, but it was manageable. Just be prepared to get your shoes dirty.