It is certainly doable. Before you buy, rent and take a few trips. Yes, rental is expensive but you need to know if you are really going to like this. And you will know what type of RV or Travel Trailer you will be comfortable with. Have you ever towed anything? There is some learning curve, particularly in backing up. Then if you decide on a TT you need to match up the tow specs to your tow vehicle and see if you need a new one of those - or perhaps then you will choose an RV instead of a TT. You might like a Class B RV - big enough for 2 and can go anywhere a car can go.
Think of this as a whole new way of traveling. Many love it - some don't. It took us a year to say that we loved it - and we still do. We go sightseeing and not camping. We are out in the morning to see the sights and come back at the end of the day. Others love to stay in the campground and just look at nature and the other RVs parked around them. We will never stay in another motel/hotel again. There is a great benefit to sleeping in your own bed with your own sheets and not dealing with anything that the last guest left behind in the room. Is there a drawback to this - well if there is something wrong with a hotel room you ask for another room - if something goes wrong in your RV you stop your trip until you have it fixed or fix it yourself. As I say, it takes some getting used to.
As to expenses on a trip without factoring in the initial purchase costs and insurance - this varies by parts of the country. If you come to the East Coast you will encounter campgrounds that charge from $35 to $90 and more. We also discovered that there are not always campgrounds near to where large attractions are and cities can be a problem in general. The best way to find out is plan a fantasy trip - check out the campgrounds and set up a trip you might take Here is a link that will show you campgrounds for any location you put on the map on this link -
http://www.epgsoft.com/CampgroundMap/ For each campground there will be a link to reviews and the campgrounds website for current rates. Then compare that to what the trip would cost if you stayed in motels and drove your car. Expect that gas will be twice or more than what many cars will use. Food can be cheaper if you don't mind cooking every night. We use restaurants - it is a vacation for both of us.
Check out my site linked below to see what it was and is like for us who never owned an RV before. Go back to the beginning to see what we experienced shopping, buying, etc.