Contact fellow forum member "Eagle." I had inquired about such a trip and he gave me some solid insight. I've had that PM saved for a little over two years now. I think he used to work there...maybe still does. Good luck!
Thanks Finch! I worked as a wrangler in the park in 2007 at Roosevelt fwiw.
As for recommendations, with the two young ones along, that can change some things.
First off, if you want to stay in the park at any of the lodges, start looking at availability yesterday. These places book quick, generally a year in advance. I would advise on going in late August since the kids aren't in school, so long as your job allows. The weather will be more stable, mosquitoes shouldn't be too bad, and the crowds will be much less severe.
As for the park itself, you can stay at Lake Yellowstone, Old Faithful, Mammoth, Canyon and Roosevelt. Grant Village may have lodging, but I'm unsure on that. Within each location, there are some things to see. At Lake, the old Lodge is beautiful and worth checking out, as well as the Lake and Fishing Bridge area. Old Faithful's Inn is awesome and of course there's the geyser and geyser basin for which it's named. Mammoth has the terraces and in the fall, lots of elk. Canyon has the awesome Yellowstone River Canyon and the upper and lower falls. Roosevelt, my personal favorite, has paradise valley and the cowboy cookout dinner, which you and your young family can attend by covered wagon (it's an all you can eat steak dinner two miles from the lodge at a set location in Paradise Valley; accessed either by trail ride on horseback, or by one of the many covered wagons, 180-210 people are feed nightly throughout the summer from early June to early September).
The park itself has five entrances. Leading up to the southern entrance, is Grand Teton National Park, which for pure mountain scenery, is far superior to Yellowstone. The nearest town is touristy Jackson, but it's worth seeing the elk antler arches in the town square and walking through some of the beautiful art galleries. If you and the wife are up for it, doing a short hike with the kids in GTNP is worth it, if the Tram is operational, you can ride it up to the top and then hike back down to lessen the strain on the legs and lungs.
The west entrance is West Yellowstone, and as another member said, the Wolf and Grizzly Discovery Center is a must see, especially with kids. The drive into the park from West is along the Madison River and is beautiful as well.
The NW entrance is Gardiner, and this is where the famous arch is. Gardiner will take you into Mammoth, and isn't that long a drive from Mammoth if you just want to see the arch.
The NE entrance is Cooke City/Silver Gate, which is a small mountain town with a few restaurants and not much else. The entrance leads to the Tower Junction area where Roosevelt is located and takes you through the Lamar River Valley which has been called "America's Serengeti" for the amount and diversity of animal species that can be found there.
The east entrance is Cody, which is home to the Buffalo Bill Cody Museum which has the most extensive collection of firearms I've ever seen, and has an awesome art section and western history section as well. The east entrance road leads to Lake Yellowstone/Fishing Bridge area. The museum is a must see, but would likely put the kids to sleep, so not sure what to advise on it honestly.
One additional thing that I feel is worth the effort/time is to take the drive from the NE entrance along HWY 212 to Red Lodge, which takes you over the "top of world" and through the beautiful Beartooth mountains. It's an epic drive, and Red Lodge is an awesome little mountain town with good food.
That's a lot of info/places to see, and 8 days total for the road trip will make it tough to see it all. With the direction you're coming from, I'd recommend checking out Mount Rushmore as well, it won't be too far out of the way if you plan to go along I-90.
From Indiana, I'd guess you'd coming that way, so after Rushmore, you could then hit Red Lodge and travel HWY 212 to Cooke City and then down to Roosevelt. From there you could head over to Mammoth/Gardiner and then down to West Yellowstone (in the park you travel south from Mammoth before heading west to West Yellowstone). From West, you could then enter the park again and travel down to Old Faithful, then retrace your steps back to Madison Junction and from there head over to Canyon. From Canyon, you could head up to Dunraven Pass at Mount Washburn before heading south to Lake Yellowstone and then out to Cody. Retrace your steps from Cody to Lake, and then south to GTNP and Jackson before then heading out via Pinedale (the best mountain man museum I've toured is here) and joining I-80 headed east at Rock Springs on the way back home.
It'd be a ton of driving, and could prove to be entirely too much for 8 days with two little ones in the car. You may decide you'd rather concentrate on one portion of the park rather than attempt to see everything in 8 days, or better yet, make it a two week trip and take it all in at a more leisurely pace. Good luck with the planning, and if you have any specific questions, feel free to send me a PM.