Dont buy a motorhome.
Dont buy anything with a motor or transmission or that requires insurance coverage! That is for the people who want to burn some cash off.
A used trailer is fine. Heck, a $1,000 - 3,000 used trailer can be 100% functional and not too ugly at the same time.
Critical areas are 99% look for water damage. This happens mostly on the roof, but the roof leaks into the walls too.
Water damage is everything!
Look for water damage.
Ask about water damage.
Smell for smells indicating water damage (mold).
Look for water damage.
Water damage can be repaired, an maybe it was by the owner and maybe it is good to go, usually it has some level of water damage that is not so good.
Checking out the appliance is easy enough.
Stoves and heaters and water heaters almost always work, they dont tend to break a lot, but DO check for propane leaks. Turn the propane tank on first as it might take a while to be able to smell propane leak, sometimes you may never smell the leak that is leaking.
Make sure they have propane to turn on. Bring your own BBQ tank just in case.
Refrigerators often are ruined. Have then turn it on before you arrive (at least 12 hours before). Propane / 3-way fridges will be ruined if run off-level too many times. We ll have done it, and the damage will build up to where the fridge cooling unit needs to be replaced and youre looking at $600-800 just for the rebuilt unit, plus installation.
Electric only fridges are less of a worry.
A residential or dorm fridge is useless, dont buy an RV with a household walmark fridge in it. They ruined the RV fridge and stuck in a $99 fridge just to sell. These need household current all the time, so your battery and propane will not run it. If a household unit then offer $600 - $1,000 less unless that is built into the price already.
Check the plumbing - easy enough.
Look at the tires for uneven wear indicating bent axles.
Get the lightest and smallest camper you 3 can live in. Easier to tow, easier on the vehicle, cheaper on gas, easier to park, goes more places, stores easier... Big is nice, but this is a camper, not a luxury hotel. IT needs to provide you the functions, not provide a resort experience. Keep this as your perspective.
Rent an RV before you do any buying. It will be money well spent. Otherwise you will, WILL, have buyers remore on your poor choices.
There is too much to advise on. You need to ask specifics and give specific budget and specific needs,...