"hence the $6000 budget"
With this little money, I would limit my search to the least demanding vehicles price wise I could find. Focus wagons and compact wagon like hatch backs that your dog could fit in the rear would keep the reliability up. I would avoid all pricy SUVs, minivans etc. which would drain too much out of the sum to get a good car. A Kia or Hyundai compact hatchback could also get your mileage well below 100k for $6k.
Again. I would be very flexible on the type of vehicle and shop for bang for the buck. You have very little choice with $6k. Getting an SUV or minivan with 120K to 150k is no savings if it needs $2k worth of repairs in just a few years. Even a large dog can curl up comfortably in the back of a hatchback. After a couple of years, the car is still useful enough to keep as a second car if you wish to go bigger.
With your budget you might want to expand your search. The market for used CR-V’s and Rav4’s is strong because these cars are in high demand vs supply. Meaning they are more expensive to buy used.
Just how many miles per; day, week, month do you drive? There are a lot of Ford Explorers on the market and they are not in as high demand. You will spend more for gas but if you don’t drive a lot of miles the lower costs to buy and maintain could balance the higher fuel costs. Explorers are very reliable, seem to rarely need expensive repairs and are very safe and roomy for a family.
A midsize sedan will have an easy to access back seat and a large trunk and should be plenty big enough for a new family and get better mpg and costs less to buy a newer and lower mileage car.
Rav4 and CR-V are good cars, but at 150K getting a used one is a gamble. CR-V’s in particular are very sensitive to using Honda brand fluids in the front and rear differential units and transfer case. Lack of maintenance of these components or use of the wrong fluid can make for a $2000 repair bill.
One child doesn’t really mean you have to get a newer, safer, or bigger car as long as you aren’t driving a tiny Smart Car, or a Fiat 500. A 4 door civic or corolla can handle a couple with a child without any real problems.
Lots of good advice here. As others note, as mileage goes up, so do needed repairs, so total operating costs may not be such a good deal. Cars with over 100k miles used to be worn out, but now usually have enough miles left to be useful to someone like you. Unfortunately, a decade ago there were not many wagons/hatchbacks/compact SUVs around as there are now, except for European wagons, and they should be avoided for poor reliability.
The Explorer another mentioned could be a cheap option. There were a ton of them sold and they are completely unfashionable now. Nice and roomy with plenty of space for evsryone.
There are other largely forgotten SUVs like the Mitsubishi Montero (still sold, but barely) and the Isuzu Trooper 2. They aren’t so bad, but parts may be hard to find (especially for the Isuzu). Isuzu also made the smaller Rodeo, also sold as the Honda Passport. I can’t really recommend any of these because there are so few around and parts can be scarce.
Once upon a time there were wagon versions of the Camry and Accord, as well as the Corolla (and the Civic, a long, long time ago) . Nice cars, but Toyotas and Hondas always cost more, though that may be offset by these being wagons, an unpopular body style.
There also was the ubiquitous Taurus wagon. My favorite wagon was the Mazda6 wagon of about 6 years ago, but they are probably too expensive. They also sold poorly so few are around, but it was a beautiful car and a nice size. There were wagon versions of the Hyundai Elantra off and on. Not a masterpiece, the older Elantras, but not trash, either. If you want a really exotic wagon, there was a wagon version of the luxurious Mitsubishi Diamante. Very swanky and very rare. Likely to be expensive to keep running, but you’ll have the only one around. I don’t remember when I last saw one.