When buying a used Fiat Punto mk2 or mk2b, what are first few things you check before anything else ?
(things that if turn out to be bad, you walk away)
Apart from excessive rust.
I have seen more older Fiats burning oil/worn pistons than knocking bearings in bottom end, but same as buying any car, anywhere regardless of age etc. it is usually better with two of you looking, as no matter how good you are it is easier to miss something on your own.
Also seller less likely to try and B/S you!
Buying a vehicle
I always take my time, preferably with engine from cold, check under bonnet, including oil levels and smell the oil and check colour etc.. Also the colour and smell of the coolant, combustion or oil, signs of recent topping up of fluids/ spillage etc. before you arrive.
Then have a good look around the rest of the vehicle, including windscreen damage, under carpets, also under vehicle as much as is possible, not just for rust, but also any leaks of oil from brakes,gearbox engine, radiator, wheel and tyre conditions, signs of accident repairs etc.
Look for a good firm brake pedal and handbrake operation.
When finally happy about that, then start the engine with bonnet still up and foot on footbrake which should go down a little to indicate the brake servo working, listen to the engine and look for exhaust colour etc.
Then a decent long road test, checking all gears for quiet and easy engagement including reverse, along with no signs of wanting to jump out of gear.
Feel the clutch operation, firm, not sloppy, easy gear selection,if possible with engine warmed up and pulling well up a slight hill and going fairly fast as you change from say third to fourth gear I try to accelerate quickly, but be a little lazy bringing the clutch pedal up, this will give a better knowledge of how good the clutch is, rather than what I have seen others do of revving engine and trying to pull away from a stand still in fourth gear! Don't abuse it, or the owner will quite rightly tell you to f*** off

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If safe to do it, I try an emergency stop with my hands off the steering wheel, to see if vehicle stops in a straight line, you may need to warn the other occupants before doing this

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If a suitable hill try the handbrake also to see if it easily holds the vehicle. On pulling away you can check for clutch judder.
All the time whilst on road test check for smoke, engine temp. strange noises and anything else that to you doesn't feel normal.
If applicable check for Air Con operation.
On return from road test open bonnet whilst engine still running and look for any leaks, smells, strange noises etc.
Check all wheels to see if any brakes are getting unduly hot.
Finally after vehicle has been switched off for a few minutes see if it happily starts again.
Any genuine faults you find decide if it is time to walk away or point them out to the seller to negotiate a better price.
Always be prepared to walk away!
Note! However good you check you will almost always miss something.

Many years ago I had a lady customer looking for a car, we spent the entire day going around dealerships in a 30 mile radius and looked at a lot of cars, all the time I was talking to her and showing her the faults I found in the various cars.
At the end of the day I apologised for not finding a suitable car, to which she replied she was very happy and had learnt a lot along with not wasting many thousands of pounds, she paid for our lunch during the day and at the end happily paid my hourly labour rate for the entire day.
