Paul, there are a number of ways to enjoy some really scenic parts of the JMT/PCT over the course of a weekend (or especially a long weekend).
Rae Lakes Loop - start from Roads End trailhead in Kings Canyon National Park. Hike up Paradise Valley until you catch the JMT. Follow the JMT through Rae Lakes, up over Glenn Pass, and loop back to Roads End. 44 miles and maybe 7,000' of elevation gain - I did it in two very hard days, though most people take much more time than that. This can also be done from Onion Valley trailhead on the east side, and it's shorter.
Evolution Basin - My buddy and I did this over 4 days (it was roughly 65 miles). Drive to Bishop, drive up to South Lake, hike over Bishop Pass and drop down to the JMT, then hike along the JMT until you hit the Piute Canyon Trail. Hike over Piute Pass back to North Lake, then hitchhike from North Lake to South Lake (or maybe stow a mountain bike at one trailhead?). Alternatively, you can do a shorter loop starting and ending at North lake that involves travel over Lamarck Col, but do your research, as this is an off-trail pass that can be challenging to follow. Best done in September.
Lake Thomas Edison - this lake on the west side is very close to the JMT and a number of short loop hikes are possible from there, including the beautiful Silver Pass area.
Mono Divide - a gorgeous way to access the Lake Thomas Edison area from the East. It involves either climbing up from Pine Creek Canyon over Italy Pass (5,000' of gain and not much of a trail) or go in at Mosquito Flat and take an off-trail route known as Cox Col. - Not easy! I actually had a surprising amount of trouble descending to Lake Italy once up over the pass (bear north - if you head directly for the lake it cliffs out)
Lyell Canyon - this is a hike from Tuolumne Meadows. A 2-day loop is possible with Rafferty Creek. I have been trying to do it for the past month, but the first time, Highway 120 closed due to snow and the second time there were thunderstorms so I opted for the Trinity Alps instead.
Agnew Meadows, near Mammoth Lakes - haven't done it yet, but I'd like to take a short weekend trip from there sometime this summer. Let me know if you wanna come