That you should do this is the theme song of the judge on The People's Court and maybe others.
It seems to me that the only tests really worth running are the "exhaust gas in coolant" test. Is there a way to do that oneself?
And maybe the compression test? Which I could do myself but I think the seller would object to it if he was there to watch, as he woudl be. But he woudlnt' ojbect if it were done at my menchanic's. But now I'm too old to bother and would just go by how the engine sounds and runs.
What I do before I buy the car is drive it, pay attention to how the automatic trnasmission shiffs up from first to second, and so on, that's normally quiet and smooth, and how it shifts down when, at reasonable speeds, I move the shift lever down.
I let go of the steering wheel to see that it goes straight ahead, allowing for any crown in the road. (although if didn't go straight ahead, I'd figure the odds are very high an alignment will fix that)
And I look for bad wear on the tires, from bad camber or caster, toe pointing, little feathery edges on the tread, etc. Improper inflation is bad too, but it just requires proper inflation, it's llike the old tires will never get right 100%, and at worst new rims.
I check to see if it has enough pep, and a high enough maximum. Once I test drove a car that just would not go above 65. Even though that's the limit, a car that works right should go a lot faster
I used to put my finger in the exhaust pipe to check for poor combustion, but I"m told catalytic converters usually leave that area clean, even if the car has got poor compustion (and poor combusiton if it actuallyl has it, is likely to be harder to fix than the odds were before.)
I look only at cars that have good bodies and good interiors because body work and upholstery are expensive and I can't come close to doing it myself. I only buy high end models of American convertibles, and the only thing wrong with that is that many have automatic AC.
And I test all the lights and accessories, including the cruise, the heat, the AC, every feature of the radio, not because I wont' buy the car, but for price negotiation. in 6 cars, nothing has ever been bad, however!
And then I allow 1000 (in the past) and now 2000 for repairs after I buy the car**
I don't ask questions of the seller, in part because I don't want to cause him to turn into a liar, but more because I don't want to be angry at someone if I find out later he was lying. It's better if he just doesn't tell me anything.***
I don't see any point in looking at the brakes, beyond what you can see through the rims. If you find out later you need pads, get them.
OTOH, getting a mechanic requires some sort of plates for a car that no longer has them, and that seems like an enormous problem.
And you need an appointment, and you have to hope he's not interrrupted and you spend all day there.
And how much they charge depends. When I blew an engine in N. Carolina, the only covertiible I could find for sale the first day within 50 miles was a wreck, a red ford mustang, but lots of things wrong with it. Someone had recommended a mechanic, only 1/4 mile from my motel it turned out, and I asked him, How much to inspect that car? And he said, "It depends on what you want me to check". And of course it does. He didn't seem eager for the job. We never got to prices. He didnn't seem to think it was worth it, imo. (I had arrived in my nice looking LeBaron, and asked him how much to fix that. The first thing he did was check if there was exhaust in the coolant, and there was, and he said it probably wasn't worth fixing. He didn't even charge me for the test, even knowing I was from out of town and he had but one chance to make money off of me. IOW, an honest guy.)
My impression has been that just as a doctor can give you clean bill of health at your annual physica, and then you drop dead tomorrow.
In fact I've read that too many people get an annual physical and treat it like a year's guarantee. So when they start having symptoms int he middle of the year, they figure it couldn't be anything because they're still under warranty. My own mother started having a symptom only two months after her checkup, and it took her an extra two weeks to decide to see the doctor. Only two weeks and they cut it out and she was fine, but if she'd waited 10 months, maybe very different story.
So, is it worth getting a mechanic to look at it, or
**Only one car out of 7 has needed more than 100 in the first two years, and that one, if I'd paid more attention, I should have noticed did not shift to high gear (3rd or 4th) but my current car had a broken exhaust pipe and made a tremendous racket, and this one was so quiet by comparison. But even if I had noticed, it was the nadir of convertible availablity, about 1979, and I'd called Car-Match, an early version of computer shopping, in at least 15 counties holding at least 12 million people, and this was the ONLY car I was interested in. So I woudl have bought it anyhow. And I was glad I did. The transmsision repair was almost 1000 but it lasted 6 years and about 100,000 miles.
***The car in the footnote above, in 1979, I don't think I asked anything, but he volunteered that he was selling the car for his son, who moved to Kansas and didn't need a car there. Huh! Years later it occurred to me that maybe his new job provided a company car, but he didnt' say that. He also said, and I wish I remembered if this was before or after I bought it, that t he cow magnet taped to the gas line in the engine compartment was there to increase gas mileage. Let me tell you, it didn't work, and mileage was terrible, but like I say, I'd looked in 15 counties with 12 million people. I could have gone home to my mother's and looked there, but a) I had no car to get there except the one with no exhaust. b) that would only be another 400,000 people. How much could that help. Hmmm. I could have had her look, then taken the bus there (2 hours) if she found stuff, then borrowed her car to go shopping. I wish I could have shared more stuff with her but if I told her in advance I needed a car, she'd have bought me a non-convertible and registered it in my name. Maybe.