Just some suggestions for new gun owners aside from the obvious safety rules. I always inspect the gun for nicks, scratches dings and dents, I don't want a dealer by chance sending me a gun someone sent back or a shipper damaging it. Next thing I do is check all fasteners/screws ETC to make sure nothing is loose. Then I clean the entire gun specially the barrel as they generally arrive filthy. From there, (PCPs) I check the gauges and top off the air wait an hour or so and mark or write down the pressure on a piece of blue tape, stick it on the stock and put it in the safe to be checked the next day. What else do ya'll do with your new guns before you shoot them?
 
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I also like to get a good look at the bore after cleaning. If anything raises questions it's. Not a bad idea to push a pellet through and feel if it's even,or if there are places that feel tighter/looser than others. Also gives you a pellet to inspect to see how well it is engaging the rifling
 
All of what the OP said!

After the initial inspection, cleaning, pressure test and test shots to confirm proper functionality I'll note all the factory settings and shoot a baseline string of shots for later reference.

I've usually done all kinds of research, downloaded drawings, parts lists and owners manuals so I'm familiar with the airgun, its specs and know what to expect before committing to a purchase.

After that there's the break-in period, followed by pellet/slug testing and, maybe, tuning to a specific projectile.

All fun stuff in the getting to know your new airgun.

Cheers!
 
Regarding waxing the bore, here's a lengthy thread on the subject that covers types of wax, preparation, application methods, etc. For the insatiable reader.