If seeing BIG trees is on your "gotta do" list, consider a visit to Sequoia/Kings Canyon National Parks. These joined parks are maybe an hour farther south from Yosemite, in the Sierra Nevada mountains. An easy overnight trip from SF.
Sequoia NP is home to several old groves of redwoods and claims to be home to "the largest living thing on Earth" - the General Sherman tree. Kings Canyon is a valley similar to Yosemite (the naturalist John Muir reported it as just as beautiful), but not quite as crowded. It also lacks the readily-identifiable rock formations like Half Dome (which is possibly why it doesn't get the big crowds?) Be prepared for driving on twisty mountain roads, though.
Note that in August even the Sierra will probably be rather hot (temps into the 90's fahrenheit at the valley floors of Yosemite or Kings Canyon. A lot of sightseeing & hiking is a bit higher up & so a few degrees cooler, but then you start dealing with lack of oxygen (The General Sherman tree is about 7000 ft/2200 m above sea level)
Speaking of temperature - bring a jacket. When near the coast (including in San Francisco proper), there is often a very cool fog. But a few miles inland it can be sweltering.
Driving north along the coast to the Avenue of the Giants, another grove of old redwoods (including a tree with a road running through it) near the Oregon border, would be another option (although plan at least 2 if not 3 days for this side trip). Muir Woods (to the south of SF, toward Santa Cruz) would be an easy day-trip with some very big (but not quite as superlative) trees.
Another oft-overlooked historical point that might be of interest along the coast a couple of hours north of SF is Fort Ross - an old Russian trading post from the days when they came south with the Aleut kayakers to hunt otters for fur (early-mid 1800's). Most folks know the Spanish influence in the region, but forget about the Russian.
Hope you have a good trip!