Fly in to Reno. A shuttle trip between the airport and either Tahoe City or South Lake Tahoe cost round trip $70-$80 for a 70-90 minute ride. I believe there is also a shuttle into Truckee, but I'm not positive.
The entire TRT is 164 miles, so probably more than 7 days, unless you make big miles, but there's plenty of section hikes. You can walk onto the TRT from Tahoe City, but the hotels are expensive. In South Lake, you can get a motel in October in the $40-$50 range, and it's a $25-$30 cab ride to either of the Kingsbury trailheads. There's a bus that goes up to Kingsbury as well, from with it would be a couple miles walk to the TH. Cabs can pick you up or drop you off at other trailheads from South Lake, (Spooner Summit, Echo Lake,both of which have pay phones) but you're looking at an expensive ride. Echo Lake offers access to Desolation Wilderness, which would probably offer the scenery you are looking for.
From Truckee,you can get to a couple PCT trailheads via taxi. Truckee hotels are also a bit on the pricey side.
The only place that requires a permit is Desolation Wilderness, and after Oct 1, they go off quota, so it's just a matter of submitting a request.
There's also a bus that runs down Highway 395 (that I've not personally used) that could get you down to the Yosemite/Ansel Adams Wilderness area, but you'd lose some time in transit.
You could also check out this thread for 48-72 hours fly-in trips or this one where the poster used public transit to go to/from Sacramento to Seqouia-Kings Canyon, definitely the altitude and scenery you are looking for.