Public transportation in the Northwest (as elsewhere) is not at all helpful for the hiker or backpacker. There are some ways to get into the mountains, but they are few and far between and generally involve a lot of changes on small local bus lines. The one exception is Trailways which, I understand, will stop on request at Stevens Pass where you can access the Pacific Crest Trail. (No, there is no equivalent bus stop at Snoqualmie Pass--Greyhound stopped doing request stops years ago, unfortunate because the Stevens/Snoqualmie section of the PCT is one of the more spectacular.) By putting together a number of small local bus lines, you can, slowly, circle the Olympic Peninsula on US 101, but US 101 is a long, long way from any Olympic NP trailheads. Trying to incorporate public transit into any hike is both time consuming and very, very limiting.

It is possible to access part ot North Cascades National Park at Stehekin by taking Trailways from Seattle to Wenatchee, a local bus to Chelan, and the ferry up Lake Chelan to Stehekin. (The ferry ride is awesome and so, in a different way, is the Stehekin Bakery.) This might be one possibility. Or you can start at Stevens Pass (see above) and hike one of the wildest sections of the PCT from there to Stehekin.

There at least used to be a section on the Washington Trails Association website on hikes with trailheads available by bus. There's also a similar section in the Field Guide on oregonhikers.org.

Check out Rent a Wreck for used car rental. Be sure to watch for restrictions in the contract. Most trailheads are not on paved roads. It seems to me that with the fairly large number of hikes and locations, a rental car is your only option

Car prowl can be a problem, especially at trailheads close to main highways such as I-90 and I-84. Definitely leave nothing (of value or otherwise) in the car!

The torums at oregonhikers.org (Portland area and SW Washington) and nwhikers.net (Seattle area), might be of help,

Also, it would be helpful to know at what time of year you plan this trip. Mid July through mid-Septe,her is the optimal time to avoid snow on the trails.