This is really a Yosemite question, but hopefully other Yosemite people will see your question here.
Folks make that drive all the time, but in August it will be tougher. Continue checking the official park lodging website for cancellations. Did you also look at the private property options inside the Park gates? Did you read the lodging info we linked to the Top Questions about Yosemite on the Yosemite forum page?
Tenaya Lodge is the best location for visiting Mariposa Grove. It's only about ten minutes, if you arrive early enough to get a parking place! If not, you drive past the Mariposa Grove on to Wawona (another 8-10 minutes), park, and ride a bus back to Mariposa Grove.
Assuming that there are no delays at the South Entrance gate (and I think that there will be from about 9 am till 6 pm).........
...Tenaya Lodge to Glacier Point is about 1 to 1 1/4 hrs.
...Tenaya Lodge to the Valley is about 1 to 1 1/4 hrs.
...Tenaya Lodge to Tuolumne Meadows is about 2 1/2 to 2 3/4 hrs.
The most scenic route from Fish Camp to South Lake Tahoe (the city ... don't know where you're headed at Lake Tahoe) is about 225 miles and at least 5 hours of straight driving. But since this is a very scenic drive, it should take closer to 10 hours. Drive through the Park to Hwy 120 / Tioga Road and slowly savor this drive. Stop at Olmstead Point, Tenaya Lake, and Tuolumne Meadows. Outside the Tioga Pass entrance are more lakes & more scenery. North on Hwy 395, perhaps stopping at Mono Lake (tufa spires) and an additional 2 hour detour to Bodie Historical Park / Ghost Town. This is a very long day, but it's a good way to include the beauty of Tuolumne Meadows in your Yosemite tour.
One option is to leave Tenaya Lodge a day early and truly enjoy the Tioga Road drive. Then stay in tiny Lee Vining (if you can find lodging!) outside the Tioga Pass entrance. The next day visit Mono Lake & Bodie, then continue on to Lake Tahoe. In August, the "best bits" of Yosemite is in the high country.
Read the Itineraries on the Yosemite page as well as relevant threads for more ideas.