MWM, I started to go this route before. I bought a nice pop-up that I got a GREAT deal on. Then, I sold it because I was afraid that my dogs would tear the screen and possibly get loose if we left them alone in it. But, here are things to consider- the towing vehicle needs to be able to carry the weight. This can be checked online or by calling the dealer of that vehicle. You will need to know towing weight (not the same as overall weight) of the camper to be towed). Also, consider the height of the camper as it is necessary to be able to see over the top when you are in the towing vehicle. This is not a law and many people don't have that, but I found it important since I wasn't used to towing something.
If you can swing the cost of the camper and a towing vehicle, that's the bulk of it. Gas mileage is not as good when you are towing, but everything else is inexpensive. Other PITA's though are that you have to pack carefully to keep weight balanced a certain way and there's a max weight you can put in there, your choices of vacation spots are limited based on your ability to drive and camp there, a lot of work goes into prepp'ing for the trip since you cut costs by taking stuff (including cooking stuff) with you.
If you get the camper, you want to think about whether or not AC is needed and if it has one. Also, don't buy it until it has passed an inspection at a reputable camper dealership. Oh- having brakes on it is good. There will be a cost to add some things to the towing vehicle- a hitch certified to carry that weight (if it doesn't come with one) and a brake/light hook-up.
I opted for getting rid of the camper because I was able to sell it for a lot more than I paid for it and I figured even though I put very little cost into it up front, I could have gone on a couple of vacations for that amount of money. What we ended up doing is staying at KOA campground cabins. You can find locations online and find out if they are air conditioned/heated. Prices vary depending on campground but are pretty reasonable. Some allow small dogs. They are small, but so are most campers. Most campgrounds have pools and activity areas and are located near interesting places. I take a very small charcoal grill and a small tv and difficult child takes his game box. I started a "collection" of older pots/pans, utensils, etc, that I keep in a medium sized cooler for easy access and packing. They have different sized cabins with different amenities but if you plan to take the dogs (which is fun and saves costs of boarding), typically you can only get a 1 or 2 room cabin. We get the 2-room and use one for my bedroom and put the tv and cooler in it, and use the other for difficult child's room and I put the suitcases and other kitchen stuff in it. They have elecetricity but no running water. The bigger, nicer cabins have water and kitchens, I think. The ones we get have a double bed in one room and 2 bunk beds (to sleep 4) in the other room. They are small. I would recommend trying this before investing in a camper and vehicle, unless you and the kids are already used to camping. We take leads and tie the dogs to the porch railing when we are cooking out or just sitting outside and they love it.
Campgrounds are great, typically safe, places to meet other laid-back, friendly adults and kids. I have considered renting a small 1-room AC'd cabin for me and the dogs to stay in and letting difficult child take a friend and renting them an adjacent tent spot. Of course, difficult child will have to turn into a easy child before I go thru with that!