Yeah... Those tests that are taken for granted because a car 'seems' fine can reveal some pretty interesting things.
We just got rid of my sister's 2007 Toyota Corolla, it had 44K miles and 'seemed' fine. First time I saw it I heard a noise (a not normal one) but it only barely showed up at idle, and only got even a tiny bit loud between 1000-1500 rpm. No big deal right? It went to a mechanic and they claimed it was just the wrong weight oil in the car. Amazing how things are these days.
The timing chain tensioner was either defective or deprived of oil for a long time, the noise was the chain not being tensioned. As the load increased the noise went away since the chain was being pulled tighter by mechanical force. Removal of the engine front cover revealed metal in the oil there, and in the drained oil that was a month old. Something plugged an oil passage somewhere, the car was run with too little oil for to long, the oil pump gear cracked... who knows.
Anyway, It went back needing a new engine.
Tests that should always be run:
Visual inspection (I mean like 30 minutes here, not just a once over. You're looking both for problems, accident damage, and evidence the car has not been serviced on schedule) I love this part because sellers hate it... it's intimidating to them and if they're hiding something they will get nervous. Point out everything that is wrong, and why it's wrong. They really hate that... you're already dropping their asking price.
Current oil condition (might not say much, if they're hiding a problem they will have changed the oil)
Current coolant condition (full, clean - again, may have been changed or topped off)
All other fluids and their condition
Inspect spark plugs for normal wear or signs of issues (they may have changed these too, you'll know when you look)
Test drive vehicle from gentle at start-up until warm, then like you stole it. Listen for noises, clunks, shift issues, non-working gauges)
Any anomalies like new parts and fluid, ask them about it. You'll be able to tell they're lying, because chances are good they are intimidated by your knowledge of the vehicle at this point.
If all that seems decent, ask if you can either have the car inspected or sign something to take possession for a day (most sellers with proof of insurance will allow this). Compression test is good, that plus a leak-down test is better, but the compression should tell you what has been hidden. Blown head gasket, where it's blown, how badly, leaking valves, poor sealing rings, it really tells a lot. If you can squeeze in a pre-smog, that would be great too.
If it passes all that with at least a B grade (depending on price), it's probably a keeper. If you're planning to remove all of those things anyway, just make sure you have a well cared for body, frame, panels, and paint.
Use any or all of these points to help you negotiate price. Have the asking price, in hand, in cash when you are showing up to look at the car. Don't show them all of it, only whatever you are going to offer. A thick stack of $20's is better than a short stack of hundreds but throw a couple benji's in at the top.
Most people will take it, they know they need that money and it is RIGHT there.