I've not hiked an inch in SEKI but I'm planning to do day-hikes like you, itchbay. And I've done a
lot of research. So this may help ...
Lots of other recommend the Giant Forest. I'm thinking of parking at the big museum parking lot and taking the shuttle to General Sherman (will it be running when you're there?), then walking back via the Congress Trail, Circle Meadow, Tharp's Log, Log Meadow, Eagle View, Bobcat Point, Moro Rock and Hanging Rock. That must be up to about 8 miles, maybe a little less? And packed full of giant trees, lush meadows, local history, great views, etc. If the legs permit, you can walk back to the Museum parking lot or just take the shuttle back from Moro Rock.
The Big Trees/Round Meadow trail is an easy one that I've not seen others mention. Oh no, wait, chick-on mentioned it. It's also accessible from the museum parking lot. Big trees, lush meadow, pretty flat, ~1.5 miles?
A lot of folks who know a whole lot more than me say that the best day hike in Sequoia is the trail to Heather and Pear Lakes, via The Watchtower (and above Tokopah Falls, which others have mentioned). I probably won't be doing that one because it's mostly an out and back and I have a personal prejudice against out and backs ... unless someone can persuade me it's possible to scramble above Tokopah Falls to Pear Lake. Then it would be a fabulous circular day hike, I think. Hmm, are they having a big snow winter out there? Maybe this one would be too snowy in May?
Between the Giant Forest and Grant Grove, there are plenty of random bumps with trails up 'em. Others have mentioned Big Baldy (longer trail but maybe you don't have to do the full length to get the views?) and Buena Vista Peak (apparently, can be easy to inadvertently leave trail and end up too far on the west side with a cliff between you and the summit) There's also Little Baldy closer to the Giant Forest.
Near the Montecito resort, between Giant & Grant, there's a paved forest road that heads east. After a while, a relatively good unpaved side road heads north to the Buck Rock lookout. Apparently, a short steep hike, including lots of steps, take you to a killer viewpoint. If you carry on along the paved forest road several miles, it becomes unpaved around Horse Corral Meadows. If you've got the vehicle for it, a couple of extra miles takes you to Summit Meadow from where a short steep trail/scramble takes you to the top of Lookout Peak and fabulous views over Kings Canyon and all around. Again, this is high up so it might be snowed in in May.
JustKeepWalking mentioned Panorama Point near Grant Grove. From there, the Park Ridge Trail heads south ~2.5 miles to another lookout with great views. You can go one way on the trail and return on the lookout access track for variety (well, that's my plan given my, er, prejudice).
If you can make the drive all the way to Kings Canyon for a day, the Zumwalt Meadows trail is apparently short, easy and very nice. And Roaring River Falls is also quick and easy. If you drive all the way to Road's End, you can do a flat easy scenic ~4 mile loop on the valley floor further east. It's on the way to Mist Falls, which others have mentioned, so you can zip up and down there, or go up the Bubbs zigzags a short way to get a view back down Kings Canyon.
If you do any of these, itchbay, could you report back and then I can benefit from your experience to modify my plans. Thanks.