I do the same thing with a brand new firearm that I do annually with my own firearm. The procedures and chemicals have changed over the years as I've learned new tips, tricks, and techniques, but the principle is the same:
1. Strip it bare
2. Grease it up
3. Wipe it down
4. Oil it up
5. Wipe it down
Let's take these one at a time:
1. Strip it bare: Check with your manufacturer on what products to use, but the goal here it to take it back down to the bare metal. A number of different chemicals, including oil and salt from people's hands, come into contact with your firearm before you walk out of the store.
I use brake cleaner (the non-liver-destroying kind).
DO be very careful about any plastic inserts into the sights or anywhere else., and you can't use it on all firearms! I find it works just fine on mine.
Then, without any delay for oxidation...
2. Grease it up: I used to use Rusty Duck's spray-on grease. Small can, 120 wt, synthetic blue grease. Can't find the stuff any more.
These days, I just use
fully synthetic waterproof wheel bearing grease, working a dab into all moving parts and on all surfaces. Heavy weight bearing and waterproof. Can't ask for much more than that.
Word of caution: Avoid greases with lithium, molybdenum, calcium sulphonate, petroleum, and other ingredients. We're just looking to coat all metal parts with pure synthetic grease. I use "Lucas Oil 10301 Heavy-Duty Wheel Bearing Grease." All the right stuff, none of the wrong stuff, safe for plastics (important for grips), and it's very long-lasting.
3. Wipe it down: After wiping it down thoroughly, I use high-pressure air to dig the rest out of the works and hard to reach places. Using air leaves a very thin film of grease.
4. Oil it up: Remington gun oil. Just a drop or two on all moving parts. Work the action and all moving parts several times. Swap the barrel thoroughly to remove any built-up grease.
5. Wipe it down: Again, but without the air this time.
Total time: About 30 minutes.
Throughout the year, between deep cleanings, I use Hobbs #9 to clean it after each firing, then Remington Gun Oil. I oil moving parts monthly with Remington Gun Oil.