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.