There is not much to do, simply grab the planes log book. In there you have to find ARC, airworthiness certificate, registration certificate, noise certificate, radio registration certificate, ELT registration and the certificate of insurance, which will tell you the basics.
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If these documents and the POH are not in the plane, don't go flying, as in a ramp chek you are in deep ****in trouble anyways
You don't carry the engine/airframe/prop LOGS. You may not need an ELT.
We have a folder in the plane with Radio, Insurance, Permit to Fly, and a DGAC approved manual. (French build).
I'd check insurance docs very carefully before flying a privately owned aircraft. For example, what is the deductable in an accident?