From Denver, the easiest way is to come up from the South via the Tetons and into Yellowstone from the South. You aren't likely to see much of the Park on a family visit unless you spend a lot of time in a car. Yellowstone itself is bigger than Delaware and Rhode Island combined. The roads are long, and you will be restricted to a 45 mph speed limit (which is very necessary when animals virtually live in the road).
If you don't want to drive and see everything, though, you should definitely focus on the highlights. Spend some time at the Grand Canyon of the Yellowstone, for instance, or spend some time in the geyser regions. There you will find numerous smaller hikes that don't take you too far off the road but allow you to see a lot without driving too much.
If I were you, I'd start in the Tetons, drive through them (perhaps take the ferry across from Jenny Lake and enjoy the mountains). Then, I'd head up into Yellowstone. From the Madison Campground (if that's where you are staying - are you staying in lodges or campgrounds?), you are in a good location to reach almost any spot in the Park you want fairly easily.
The possibilities are endless, and you really can't go wrong (whether you drive a lot or a little). No matter what you do, you won't be able to see and do it all. I've lived in the Park during 5 summers, and I've barely scratched the surface.
Jim, we will be staying 6 nights/7 days in the park 3 at the lake, 1 at Old Faithful and 2 at Mammoth so we'll take in what we can. We are really looking forward to it. It will be early June though (6/3 - 6/9). Can I expect muddy and buggy condition then? I realize the potential for winter weather still exists but our kids end school at the end of May and we wanted to beat most of the high summer traffic. Thanks for your reply.
You should expect to still see some snow on the sides of the roads at that time, but the weather should be otherwise clear. Almost everything in the Park should be open.
It will still be a bit early for peak bug season, though the ground on some hikes can be soggy.
Several trails will probably not be open yet, either.
You will beat most of the traffic. I went last May, and despite so little of the Park being open, it was so fantastic to drive without having so much traffic.
The weather will likely be cool but not intolerable. Expect very cold nights, but since you are staying indoors, it shouldn't be a problem.
One thing you should be wary of in Yellowstone is the dramatic changes in weather. Always have rain gear nearby, even on the sunniest of days. The weather can change in a flash. I once got stuck on the boardwalks at Old Faithful on a sunny day with only a few puffy clouds on the horizon. In 30 minutes (I am not exaggerating - go here at http://www.yellowstone-online.com/images/heartcastle.html - this was taken before we got hit), we were being drenched in a cold, driving rain nowhere near shelter. Be careful of that, especially with a family...at least have an umbrella with your family at all times or a couple of plastic ponchos.