Unless there have been very recent changes that I've not heard about, Half Dome is the only day hike in the Park that requires a permit. I believe that all overnight (i.e., backpacking) trips anywhere in the Park require permits (I've yet to do any backpacking so the latter statement is not from personal experience but I'm pretty sure it's accurate).
Although you'll find many of us in this forum talking about hikes of moderate to extreme obscurity, the general consensus is that a first time visitor to Yosemite (you don't actually SAY it's your first trip but it sounds like it probably is) should do a lot of the basic things (Valley Loop, Tunnel View, one of the main "up-the-wall" hikes (e.g., Yose Falls, Four Mile Trail, Mist Trail), hell, I even went on one of those tram tours on my first visit). If you've got a car, sunrise at Glacier Point is beautiful and wonderfully peaceful (i.e., devoid of crowds) and then take a ride down to the Wawona area (southernmost portion of the park) and see the Mariposa Grove.
If you're very early in September, you'll probably still hit crowds but after Labor Day, things should calm down...especially if you go mid-week (my wife and I will be in the Park for a few days before and a few days after Labor Day so I certainly HOPE things will be calming down by then (although we'll mostly be in areas outside the Valley)).
Scan through the other topics in this Forum. There have been a number of threads asking the "what should I do?" question over the last month or two.
Enjoy your trip, whatever you do!
