You can visit the area on a day trip from Seattle, but you'll want an early start, because it can take 2 1/2 to 3 hours of driving just to get to the area. That's five to six hours right there. We always recommend taking the Bainbridge Island Ferry, especially if you are staying downtown. The alternative is to take the Tacoma Narrows Bridge route. A ferry departs roughly every 50 minutes during the day and takes 35 minutes to cross Puget Sound. The overall time will be about the same, but the ferry ride does save car time, and it gives a real sense of arriving in the Pacific Northwest.
With only a day to explore, you'll want to start with Hurricane Ridge. The views of the Olympic Mountains are spectacular. In the shoulder season it is a good idea to check the weather forecast and the ridge webcam before heading out. Depending on how ambitious you are, also check out Hurricane Hill for a fantastic walk along the ridge, or, if you are a hardy sort, take the steep climb up to Klahane Ridge. These are both amazing hikes.
If you are less of a hiker and more of a driver, you might be able to fit in a visit to Lake Crescent and Marymere Falls. Alternately, you might take Kitchen Dick Road down to Dungeness Spit and take a walk out into the Strait of San Juan de Fuca.