I agree that without wheels, you will be severely limited as to where you can go. You'll either have to hitchhike (dangerous) or go with hiking clubs which carpool (most of their trips are dayhiking). One big complaint in Oregon is that even in the Portland area, hardly any trailheads are accessible via public transport. As Tom mentions, Yosemite would be far more accessible for you. The scenery there is awesome and, once you're away from the popular areas, it won't be overcrowded. You will need permits for overnight camping and you'll need to store your food in a bear canister. The Yosemite NP website (nps.gov) will give you lots of ideas.

While we have lots of awesome scenery in Oregon, there is more of it in Yosemite, IMHO. I don't feel slighted in the least!

You may have more altitude issues in the Sierra. Try to pick a first trip where you gain elevation gradually, and plan to take short days. After the first week, you shouldn't have any problems.

Again, June is really far too early down there, too. Try starting mid-July at the very earliest. Not only will June feature snow-covered trails (requiring knowledge of snow travel, such as using an ice axe for self arrest, and navigation skills), but it will also feature high water and hazardous stream fords.

I know it's not the West, but the southern end of the Appalachian Trail would certainly be better for early summer, would be closer to home (maybe your parents could do a drop-off/pick-up on a couple of weekends?) and would have more opportunity than Oregon for you without wheels.