For only two days, spend as much time as possible in Yosemite Valley. Start by getting some maps of Yosemite and start familiarizing yourself with the layout of the park. Come up with a game plan on what you want to do. You don't have much time. Don’t just show up. That is what I did on my first trip.
You can ride your bikes around the Yosemite Valley floor. From what I remember, bikes are not allowed on the hiking trails in the park. The valley floor stretches for roughly seven miles and is very bike friendly. They even rent bikes near Curry Village. Bikes are a perfect way to get around and you’ll also get a good feel for the valley.
Here is a brief list of some hikes to check out in order.
1. Mist Trail (7 miles, 2,000 AEG) - This hike takes you past Vernal Falls and Nevada Falls. This is also the lower portion of the Half Dome hike. It's the most popular hike in the park. You hike along the Merced River as you gain elevation to the top of both waterfalls. There is some nice swimming above Vernal Falls as long as the river is not flowing too strong. It should be light in mid-August. http://hikearizona.com/photoset.php?ID=11449
2. Upper Yosemite Falls (9 miles, 3,000 AEG) - This is another popular hike. You gain a lot of elevation as you work your way up the trail. There are a bunch of spectacular views of the falls as well as Half Dome and Yosemite Valley. I recommend heading all the way up to Yosemite Point. It's roughly a mile past the top of the falls and well worth it! This is one of my favorite hikes I've ever done! http://hikearizona.com/photoset.php?ID=11397
3. Lower Yosemite Falls (Less then 1 mile and its paved) - If your in the area and have time, make a quick detour over to the lower portion. Bridalveil Falls are similar with length and time.
4. Glacier Point - Absolutely spectacular views! This is possibly my favorite place on earth. You can hike up from the valley floor (Four Mile Trail or Pohono Trail) or drive your car. It's worth the time getting there. http://hikearizona.com/photo.php?ZIP=198832
Try to hike over to Sentinel Dome if you drive. It's a short 2 mile round trip hike from the road.
5. Some sites to check out if you can...Olmsted Point, Tunnel View, Ahwahnee Hotel and Mariposa Grove.
I could go on and on but I want to keep this brief. You really can't go wrong with whatever you do. And yeah the crowds can be pain but don't let that discourage you.