Crystal Cave is on the eastern slope of the Sierra Nevada, about an hour and a half drive from Three Rivers. The cave is one of the many in the Sierra Nevada that are made from marble. You can buy tickets for a cave tour from the Ash Mountain/Foothills or the Lodgepole Visitor Centers. While the cave is quite beautiful, it is not the best I've visited (you'll find better in Virginia), and not as interesting as most of the park's other attractions. The tour is $11 per person.
Kings Canyon National Park is a bit far from Three Rivers, but the good scenery is worth the drive. From Three Rivers, it's almost a two-hour drive on Generals Highway through Sequoia National Park to Grant Grove, where it's still an hour away from Cedar Grove, the heart of Kings Canyon. Kings Canyon is the deepest canyon in the US, at over 8,000 feet deep (the Grand Canyon is a little over 5,000 feet, and Hell's Canyon in Oregon is 6,500 feet). The Kings Canyon Road itself is scenic, but getting out and hiking is rewarding. Trails lead to waterfalls (Mist, Roaring River, Grizzly) and meadows. While it's a good day trip, it's a better overnight stay.
Lake Kaweah,a reservoir, isn't too attractive, though it is close to Three Rivers. On a weekend, you'll find many people boating on this large lake. From the western edge of the lake, you get a great view across the reservoir to Alta Peak, one of the most prominent peaks in Sequoia. If you enlargen my photo you can see it.
Do the foothills look bleak and unattractive? Think again... although they are not as sublime as the high peaks of the Sierra, they have their own beauty, ranging from canyons and waterfalls to Native American artifacts. The foothills are often hot and dry in summer, but temperatures are usually more moderate in spring and autumn. The main visitor center for Sequoia National Park is the Ash Mountain/Foothills Visitor Center just 5 miles from Three Rivers. Also in the foothills are the Kaweah River, Hospital Rock, and the Sequoia National Park entrance sign. So while many other visitors would just pass by this area, the foothills of the Sierra still merit a stop.
Mineral King Valley is one of my favorite places on earth; it is a tiny subalpine hidden in Sequoia National Park, 25 miles from Three Rivers via a winding and narrow road. The scenery of Mineral King justifies the drive, though; the road dead-ends in the end of the valley, where you can hike from green meadows up to lakes hidden in tarns. Mineral King is the only place in Sequoia National Park that provides direct access to the Great Western Divide. Some of the more prominent features of the valley are Farewell Gap and Sawtooth Peak, both of which have been photographed many times. Good trails in the area include a 7-mile round trip hike to Eagle Lake, and the short Cold Springs Trail.
Giant Forest is home to the largest trees on earth, the Giant Sequoias. This is obviously the highlight of Sequoia National Park. The largest living thing on earth, General Sherman Tree is in northern Giant Forest and an hour from Three Rivers. There are many miles of hiking trails that pierce the wilderness of Giant Forest; easier and more popular ones include Congress Trail, Big Trees Trail, and Hazelwood Nature Trail. Not to be missed is the informative Giant Forest Museum, which provides information and exhibits on sequoias. Not far away is Beetle Rock, a granite dome with views of the foothills. From Generals Highway, you should consider driving the Moro Rock-Crescent Meadow Road, which has attractions like Tunnel Log, Moro Rock, Crescent Meadow, and Tharps Log.