Hi - went there twice, once as a student in July-August '93 and back in my 80 with two small kids (2 yo and 6 mo) in May '08. First of all, I got snow each time

! So pack some heat, hem, warm clothes.
Like the others said, this place can be the backdrop of one of the trips of your lifetime, but summers are indeed very crowded. Queuing for half an hour just to find a parking space near the geysers/bison herds kills a lot of the fun.
So since you might have the choice to go there in September, just do it and enjoy.
With kids (and strollers?), there's still plenty of fun things to do, on top of my list are Old Faithful, Lake Area and the Grand Canyon of the Yellowstone. Remember that tourists are often shamelessly lazy, so walking 10 minutes often makes the pedestrian traffic disappear.
Should you want to try something more challenging with kids walking along or in a backpack-type carrier, the river trail next to the Yellowstone river heading north to Gardiner is a fond memory.
As far as accomodation is concerned, and unless you're loaded and can afford the luxurious Lake Hotel or are camping (remember, altitude is roughly 6,000 ft and it's cold even in Summer), I'd advise you to spend your nights in West Yellowstone, there are plenty of cheaper motels and nice restaurants to dine in. Some might say that "commuting" every day to your destination of choice will be a pain, but I beg to differ as the landscape and wildlife changes a lot from one day to another. And kids love bisons/coyotes/birds!
As far as wheeling, there are no trails whatsoever in YNP, but the forest in the NW of the park (whose name I don't remember) has lots of them. The Grand Teton NP has a "river trail" but I was told that this one isn't exactly challenging.
Anyway, enjoy and take pics!
