Staying in hotels/motels with a dog can be tough. Our last dog was a small (5 lb.) Yorkie, very lovable, well manicured and well trained, but most hotels don't allow pets in the rooms. Those that do often charge an extra $10.
I often carried our Yorkie to the check-in counter so the desk clerks could see her. Sometimes they'd tell me to "sneak" her in to the room. "Just don't make it obvious, and she'll be fine." And sometimes I'd sneak her right past the check-in counter, tucked away in my pocket, a luggage bag, or hidden in rolled up jacket. But no matter how we did it, traveling with the little pooch was a chore. A bigger dog would only be harder.
Compare that to a dog in your RV. No problem. I'd guess that half the campers I see have a dog. We've stayed at a few RV parks that charged an extra $5 per pet, but they're in the minority, and even with the extra charge, the campsite price is still pretty reasonable.
If your dog had a vote, (s)he'd vote to go by RV.
As for traveling, if you're in no particular hurry an RV can be a wonderful way to go. And it can be inexpensive. If you or your spouse is at least 62 you can get a Senior Pass for free admittance to all the national parks and monuments. It also gets you a 50% discount at national forest campgrounds and, I think, at COE campgrounds. Where we live that amounts to $6/night camping. We don't get hookups for that price, but we prefer the woods to concrete and manicured grass -- that's why we have a self-contained camper.
If RVing is new to you, I'd join the others in suggesting that you rent before you buy, even if it's for a few days at a time. A class B would be ideal for traveling in, but if you want your camper to be your home for weeks or months on end, it could get cramped in a hurry. I'd suggest that you rent a class B, a class C and a class A. Visit a few dealers and do a lot of looking. Talk to fellow campers about their rigs. You'll eventually decide what's best for you.
Good luck, and have fun!