There are a bajillion (conservatively speaking) trails to hike in the Sierras.
National Parks (especially Yosemite) carry with them the problem of permitting and restrictive camping/food storage rules.
One of my hiking group's favorite beginner loops is Dinkey Lakes, in the Sierra NF between Yosemite and SEKI, accessible by heading up Hwy 168 out of the Fresno area. Nine mile loop through subalpine lakes, excellent fishing (brook, rainbow and golden trout), wildflowers til fall, streams, peaks (class 2 and 3, Dogtooth and the Three Sisters are fairly easy), and no bear canister requirement (counterbalancing is still valid, tho they will highly recommend the canister as not everyone is good at hanging bags). The scenery is not as spectacular as Yosemite Valley or Kings, but still a gorgeous area. And the trail is not terrifically strenuous. You can access the loop from two trailheads, one on a rougher dirt road and the other on a longer paved route to a trailhead near Courtright Reservoir.
Ansel Adams Wilderness (adjoins Yosemite to the southeast) is also classic Sierra scenery and a great place to plan any number of loops or out and backs. We overnighted last year at Cora Lakes and could see the back of the Minarets (which we saw from the other side from the JMT). If we'd kept going we could have crossed Isberg pass into Yosemite.
Trails in the Kaiser Wilderness (not far from Dinkey Wilderness) are also good – there's a 15 mile loop that includes Kaiser Peak. We did that one as an overnight.
I would consider calling the Sierra NF rangers around mid-June to find out if these are accessible – I'm betting they will be, and probably will have one or more overnights on my schedule for August and September in these areas, after the bugs have dwindled.