What would be the ideal route to buying a gently used Infiniti QX80? What sort of problems should I watch out for?

have you bought a used car before? then same rules apply.

the question you should ask is what is the HISTORY of the car. read the report on it if you can. if private purchase, ask the owner specific questions.

i would recommend you buy CPO from nissan/infiniti and not 3rd party lot or other car brand.

the car is solid, but depends on how well it was taken. in short,. you should not find ANY UNIQUE issues. typical stuff (brakes, hoses, maintenance, etc)