You may encounter deer anywhere in the state, but the most reliable place we know is Hurricane Ridge in Olympic National Park in late afternoon in mid-summer. Elk are often spotted by backpackers in wilderness river valleys deep in the park. A better place for the average motorist to see elk is in the Toutle River Valley at observation areas along SR 504 on your way to Mount St. Helens.
Mountain Goats are less often seen. But Rainier Park is a good area to begin looking, especially on hiking trails in the Sunrise vicinity on the park’s east side (try Mt. Fremont Trail). Hiking trails along SR 20 (North Cascades Highway), such as Maple Pass and Cutthroat Pass, also may bring goat sightings. Olympic National Park is famous for goats in the Lake Constance area, and near Hurricane Ridge.
For the best mix of different kinds of wildlife, experts agree that the place to go is Eastern Washington. Without forests to hide animals, wildlife is easier to see. Here are elk, deer, coyote, red-tailed hawks, bald and golden eagles, and other raptors, and migratory waterfowl. And rattlesnakes! Start at State Wildlife Recreation Areas (Colockum, Tucannon, L.T. Murray and others) and federal Wildlife Refuges (Turnbull, Saddle Mountain, Columbia)
Link here to birdwatching site.
Link here to whalewatching site.
Link here to wildlife parks and zoos.