Hey there! I would normally agree that a rental car would be the way to go, as you could make the trip in just a few hours and have the freedom to explore the Olympic peninsula at your leisure. However, seeing as you’re doing a backpacking excursion and the car would (I assume) be sitting for days, that does seem like a waste. I did a Google and discovered that there are 2 Amtrak trains a day running from King St. Station in downtown Seattle to Port Angeles, departing at 12:35pm or 7:55pm. It takes about 3.5 hours and costs about $40. You can take the Light Rail from SeaTac airport to the train station for just few bucks. From PA, you can take Clallam Transit route 14 to Forks which runs about every 2 hours on weekdays. The ride is only 1.5 hours but It doesn’t appear to run on Sundays, and schedule is limited on Saturday. Hope this helps!