There isn't a huge number of great trailheads on the west side. You can start from Mineral King and get into trouble quickly. Plus, it is kind of hard on your vehicle to get there to start. I found it to be decent.
You can start from south of Giant Forest and go east on the High Sierra Trail. There you will have a moderate day before you get into trouble.
You can start from Lodgepole, head north, then east over Silliman Pass. I don't remember that as being tough.
You can start from near Montecito-Sequoia and drive east. There are a number of horse trailheads in there. You can get to Roaring River and back.
We will skip the Roads End and Cedar Grove trails.
You can start from Wishon Reservoir (outside the park) and go east, then you have the choice of entering Kings Canyon or stay in the Blackcap area.
For that matter, you can start from Florence Lake (outside) and go east to the Kings Canyon border in a day.
Any farther north, and you won't be hitting Sequoia or Kings Canyon.