It's too bad your folks aren't thrilled with Alaska. It probably offers the best chance to go from the airport to wilderness with little effort. For example, Juneau has several great hikes (http://www.theatlantic.com/magazine/archive/1997/02/scaling-alaska-apos-s-heights/6242/) that are accessible right from town, or at worse, a bus. The drawback of Juneau, however, is that the weather is probably not going to be that good that time of year.
The Pacific Northwest, however, will probably be very good. The best time to be here is late July, early August, but late August is very good as well (http://www.climate-charts.com/USA-Stations/WA/WA458089.php). Likewise for most of the mountain west. I would not go to the Southwest that time of year. There is gorgeous country down there, but in my opinion, it is best enjoyed in the Spring (fewer people and cooler weather).
You mentioned Portland. Portland is a fine city. It is Seattle's equal in many ways, and perhaps better in others (I wouldn't say it has better beer, but that could be a long discussion). I can understand why folks might prefer Portland for purely city reasons (some prefer apples over oranges). However, for hiking opportunities, Seattle is clearly better. Portland has basically two great things going for it: Mount Hood and the Columbia Gorge. The Gorge is really nice and can be enjoyed very easily all year round. However, I wouldn't consider it great (unless you are a wind surfer). It reminds me of the Bay area hiking trails (which are also great, but not as great as the Sierras). Mount Hood is a very fine place. It is a big volcano with lots of great hiking places. However, I don't think it is as good as the Mount Rainier area. That being the case, it wouldn't rank in the top five areas in Washington state. I would rank the (broadly defined) areas like so: Glacier Peak, North Cascades (north of Glacier Peak), Olympics, Rainier, Alpine Lakes, Mount Hood, Mount Adams, Goat Rocks. Just about everyone would argue about that order, but hardly anyone would put Mount Hood at the top. To put it plainly, Seattle is probably the best combination of a big city with big hiking. Salt Lake City is in the same league. Just about everyone else (in the U.S.) is either significantly smaller, or is further away from the good stuff.
The tricky part with all of this is the transportation. Depending on your ticket, you may be able to just land in a small airport. If that is the case, then you should definitely consider it. A lot of smaller cities are close to the great places, and fairly cheap transportation can be arranged (and the hitch hiking can be easier).
Can you go to Canada? If so, then Calgary, followed by a bus to Banff or Lake Louise might be really good (neither town is great in itself, but they are surrounded by world class hiking).
If you decide to go to Seattle (or would like to consider it) I suggest you get onto nwhikers.net. They have a ride sharing section as well as a more generic forum where you can just re-post the original suggestion. Lots of folks there know how to manage the public transportation situation, so they should be of great help. I would imagine the easiest hike from a public transportation standpoint would be to hike from Snoqualmie Pass to Stevens Pass (either direction). This is an outstanding hike that is as good as anything anywhere (especially if you go on side trips). The only possible negative is that it might have more people than some of the other hikes, but you still won't have too many.