Olympic National Park is a gorgeous 1,400 sq mi park with a lot to offer. Pets are only allowed in a few areas of Olympic National Park; however, pets are allowed in all of the 981 sq mi Olympic National Forest, which borders some of Olympic National Park.
Pets are allowed in these areas of Olympic National Park (on a 6' leash):
In campgrounds, picnic areas, parking lots
On Spruce Railroad Trail (Olympic Discovery Trail)
On Peabody Creek Trail
On Madison Falls Trail
On All Kalaloch Beaches (from Ruby Beach south to South Beach)
On Rialto Beach, 0.8 mile north to Ellen Creek only (day use only)
One of the best places to visit in the Olympic National Forest is the Quinault Rainforest, one of the few rainforests in the US. See below for more information on some of these beaches and trails: