Some of us are RV'ers because of our pets.
I'm not really clear on what you mean by a pet friendly road trip. As far as I'm concerned, all trips are pet friendly or they just don't happen. Because we don't leave home w/o our dogs. We have never found having them along to be a problem at all. In all the years we've been RV'ing we've encountered exactly one campground that didn't allow pets. So it's not exactly a huge issue. As others have said, national parks usually have pet restrictions, so that's something you'll need to consider. When we visit national parks we typically stay at a campground close to but outside of the park and do half day visits. If we're going to be gone for longer than five or six hours and the weather is nice we bring the dogs with us. If not, they don't mind staying in the comfort of the RV any more than they mind hanging out at our house.