Hi Dave
Maintenance on the Gamo Compact is easy, when fully opened there is a tiny hole in the cylinder body, a TINY drip of oil in there once in a blue moon will maintain the main seal... the pivot pins are very easy to drip a small amount of oil on to in order to keep them free
I've owned a few, and periodically the valve may stick, usually in the open position, which means you cannot 'pump' or prime the cylinder - removing the whole valve assembly to lubricate the seals (and polish up the valve stem while you have it apart) is not a massive job, just fiddly.. the tiny drop of oil in the main cylinder described above should allow enough oil (a really tiny, minute, atomised amount) through to the seals in the valve and prevent this
I know Crosman used to do very light oil for lubrication their pneumatic and Co2 pistols - not sure what others on here would recommend, but I got a small bottle of silicon oil intended for running machine belts and use that
I wouldn't put oil of any kind inside the barrel unless you were storing it indefinitely..... you'd need to get all of it our before using pistol again otherwise it will be innacurate at best, and at worst will diesel horribly