Hi Jenzie and welcome to the forum. I applaud your willingness to make a leap and try something that is out of your comfort zone. There are lots of folks here with tons of experience. That said, they sometimes also tend to think that the way they did it is the only way. That often leads to responses that might drive away new RVers from the hobby. If you don't have a 3/4 ton truck with a new fifth wheel you aren't doing it right.
I understand that you have a budget as you are still raising a family and have expenses with that. I don't know a lot about your truck/SUV but a towing capacity of 6,000 will be limiting what your choices are. Dry weights are OK for comparison purposes but most will add 1-2,000 # with options and camping gear along with batteries and heating fuel. The biggest limiting factor is your budget of $4,000. Look on Craigslist in your area and see how much trailer that will buy you. I think a trailer is the least expensive way to get into camping. Also an older trailer most likely will be heavier than the newer lightweight models.
The biggest enemy of RVs is water leaks. A water damaged RV will be miserable to camp in and expensive to repair. Rubber roofed trailer have a life of around 12-20 years if maintained properly. Check the seams and where stuff pokes through the roof to see if the prior owners kept up on roof maintenance.
The next thing is the trailer chassis. Tires are usually only good for about 5-7 years. Check the sidewalls for cracking and date codes. Then check the brakes and bearings. They should be inspected annually. Batteries will last 3 years generally with little maintenance and 5-7 years if they were looked after.
Finally there are the systems. Fresh water tanks, black and gray tanks should all be checked for leaks and proper operation. Most RVs come with AC and a furnace which are generally pretty trouble free. The refrigerators generally will last 10+ years but replacement is expensive. Hot water tanks usually have a sacrificial zinc that needs annual replacement (cheap). Then check out the propane system for leaks and proper adjustments. Finally there is the electrical system, both 12V and 120V, usually using a converter.
As you can see, there are lots of things that can go wrong on an older trailer.
Our intro to RVing didn't involve our girls as they were grown when we started. We bought a new Fleetwood Pioneer 19T4 trailer for around $12,000 (this would be too small for a family of 4). I know we made a down payment but I don't remember how much. Our monthly payment was about $126/month. We towed with our '99 Mountaineer V8 which didn't prove to be a very good tow vehicle. We upgraded to a 1999 F250 Supercab with a V10 which was an excellent tow vehicle. The difference between what I sold the Mountaineer for and bought the F250 for wasn't huge. We had amazing adventures in our little TT and explored many campgrounds in our area.
We kept that TT for 4 years and sold it for $9,000 which was less than what we owed on it. The experience showed us what we desired in our next RV. We bought a new Cougar 243RKS (could work for a family of 4) which had everything we desired. We loved that TT and kept it for 6 years. Our payment was about $145/month on the new TT. Buying new eliminates the expenses of deferred maintenance that you will find on almost any used RV. Not everyone can afford to pay cash and for the fun we had, I think it was money well spent. We sold it 3 years ago for $11,000 which was about $4,000 more than we owed on it. Sorry this was so long but I guess I had a lot to say. Good luck and best wishes. Feel free to PM me if you have any questions that you don't want to ask here on the forum.
I am NOT a mechanic although I do play one in my garage!