I take my annual vacation to Yellowstone every fall and have been there twice at the beginning of Oct. and also in Nov. I choose Oct., because that is when the hotel rates drop quite a bit. Park roads are open to vehicular traffic until early November. They close the first Monday after Daylight Savings Time, so this year it should be 11/3/08. Your admission to the park is good for 7 days and can be used in the Tetons as well.
Crowds are not too bad by early Oct., so I would expect that you wouldn't have too much trouble the end of Sept. You might experience animal jams from time to time when bison or elk cross the road.
This is the link to a moderaely priced hotel (Gray Wolf Inn) where I like to stay in West Yellowstone, MT. It is a small town but most things are still open through early Oct. and the park entrance is at the edge fo town. Visit the reservation calendar to see what the rates are. Generally they drop around Columbus Day because one of the main park roads closes for the winter. The other hotel I have stayed at is called The Yellowstone Park Hotel. It open in 07/07 so everything was nice and new. Both hotels have pools (good for your kids maybe) and free breakfast. Other towns with plenty of accommodations include, Gardiner, MT and Jackson, WY (perfect if you plan to hit the Tetons too.)
The weather starts getting unpredictable right after Labor Day, so plan to dress in layers, but for the most part you should have decent weather still in Sept. This past fall I arrived on Oct. 2 to lots of snow and temps in the upper teens. By the 7th, the temps were in the 60s and it was just beautiful. You might see snow, but who knows. They do a great job of clearing the roads though.
There are tons of places to hike and plenty of borad walks that will be easier of the kids in most places. Pay attention to all regualtions and warnings before hiking and stay on the trails and boardwalks. The earth's crust by some geysers and thermal features is not solid, so keep the kids close by. Consider a stroller or backpack for the real little ones. Be sure you have good walking shoes/hiking boots.
Plan to do a lot of driving. There are plenty of places to get out of the car and stretch or see something, but you will travel long distances to get from one place to another. Have games and other things that will entertain the kids in the car. There are some places to eat in the park, but bring plenty of snacks and drinks. We typically stock the cooler at one of the big cities along the way, usually Idaho Falls, and eat sandwiches because food is a bit expensive and there isn't a lot to choose from in the park . There are gas stations in the park, but fill the tank before you enter, it is much cheaper outside the park. Gardiner, MT has had the best prices the last couple years.
If you are looking for wildlife, bring binoculrs (even pairs for the kids) and a spotting scpe if you have one.
Big Horn Sheep - near Gardiner, MT
Otters - along the river in Hayden Valley
Pronghorn - near Gardiner, MT, Lamar Valley and in the Tetons
Wolves - mostly seen near Lamar Valley and Hayden Valley, but you might see one anywhere, check Slough Creek Campground too
Billy Goats - on the cliffs near the Northeast Entrance
Moose - near the Northeast Entrance, Pebble Creek Campground area early morning or at dusk, Yellowstone Lake and the South Entrance, but you have a better chance of seeing them in the Tetons at Moose Junction
Bears (Black or Grizzly) - the best place is Dunraven Pass, but it may be closed due to snow, otherwise it closes on Columbus Day, also try Lamar and Haden Valley (I've seen plenty in Lamar), you might see them along the road anywhere
Elk, Bison, Coyotes - everywhere
If you see a crowd along the road, there is almost always an animal to look at. If you are looking for an animal you haven't seen yet, ask other visitors or wolf and bear spotters, they can point you in the right direction and are willing to share more info than many of the park rangers. Elk are in rutt by early Oct., so be careful around them. You ought to be able to see or hear them bugling by the end of Sept....it is truly interesting and impressive.
Some other attractions:
Artist Paint Pots
Norris, Upper and Lower Geyser Basins - mostly boardwalks
Grand Canyon of the Yellowstone
Old Faithful - goes off about every 90 minutes-check at the gift shops in Old Faithful Village for the next erruption time, nice relatively flat geyser walk starting at Old Faithful, take the tour of Old Faithful Lodge or at least go inside to see it
Terraces at Mammoth Hot Springs
Visitor's center and Map Room at Mommoth Hot Springs
Caynon visitor's center - the best display and the best restrooms
Yellowstone Lake - there are water activities to do, but storms blow up quickly near the lake and I wouldn't advise this for the kids
Check with park rangers or visitor's centers to find out about ranger led activities and talks or activities for the kids. There is plenty to do, you just have to ask. You can also try this site, the concessionaire for the park.
Plan to take loads of pictures. Have enough batteries, film, memory cards, etc.
Read the newspaper you are given when you pay your admission. It has all sorts of helpful info about park rules and where various things are.
If you are in the park after dark, be very careful driving, the animals are always in the middle of the road.
Take a look at Yellowstone now. Yellowstone webcam link:
Sometimes you can see Old Faithful goo off, or you might catch bison or elk in some of the pictures.
You will have an awesome time.