Let the road be your guide
Welcome to the RTA Forum, This kind of trip is my personal favorite! 3 Weeks with a general direction, but no specific demands.
There are plenty of things in the Northwest that I'm sure you will enjoy. Certainly, spending some time in Yellowstone or Glacier could be a part of your trip. There's also lots of places to hike and explore in Washington and Oregon. Check out this recent thread for some great ideas in the area.
Most importantly, keep your trip flexable. Get some ideas of the places you want to go, but don't tie yourself down with a specific itenerary. Give your kids lots of say in where you're going and how long you spend there, but if you're having more or less fun than you anticipated, then adjust accordingly.
Oh, you're gonna have so much fun!
Greetings, Angie, and welcome to the forum!
I only have time right now to give you a list of highlights. You won't have time to do all of thes but, hopefully, this will give you some ideas. I put them in order, favorites first. You will have to keep in mind that some of these areas are national parks and won't allow your dog on the trails. But there are usually national forest or BLM lands close-by where your dog can go.
Idaho:
Silverwood Theme Park
Craters of the Moon
Payette Forest
Nez Perce Indian Reservation
Oregon:
Crater Lake
Coastal Highway
Columbia River Gorge
Mt. Hood
Bend
Washington:
Mt. St. Helens
Olympic Peninsula and beaches
Mt. Rainier
Grand Coulee Dam and the general coulee areas
Long Beach Peninsula and beaches
North Cascades Highway
Methow Valley
San Juan Islands and the various peninsulas in Puget Sound
Yakima Valley wine country
If something intrigues you and you want more details, please let me know.
Enjoy!