Welcome aboard the RoadTrip America Forums!
I'm a bit confused about your timeline, but on the assumption that you are basically taking the month of July for the trip... The drive should be made in as little as three days of very hard (600 mi/day) driving, so with 10 times that amount of time, you can wander quite a bit. Since this is your first cross-country drive, I would think that you might like to sample a good variety of what America has to offer. Keeping in mind that it might be a bit difficult to see a lot of museums and the like while travelling with your dog, you could nevertheless see a lot of natural beauty. So here, in broad outline, is just one of the many meandering paths that will let you see some of that while getting you (eventually) to Denver.
Start out by going south down the Delmarva Peninsula for one last look at the Atlantic, at Delaware Seashore State Park and Chincoteague Island National Wildlife Refuge, before leaving it behind. Cross over to Norfolk on the Bay Bridge-Tunnel and head west across southern Virginia to hook up with the Blue Ridge Parkway somewhere around Lexington and then follow that route southwest to the Asheville, NC area. Turning northwest would then take you through Great Smoky Mountains National Park and the Cumberland Gap into Kentucky Bluegrass country. Then follow the Ohio River more or less all the way down to the Land Between the Lakes, after which you could follow the Great River Road south into Arkansas. Crossing central Arkansas would bring you to the Ozarks and the Talimena Scenic Drive into Oklahoma. You would then turn west northwest through real Cowboy-and-Indian country and places like Dodge City on the way to Pueblo, CO. You could then head north to Denver with the Rockies on your left by way of introduction to your new home. As I say, that's just one possibility.
On a more general note, here are several links to tips on travelling with pets, and a primer on budgeting that you may find useful as you plan the rip that is just right for you.
AZBuck