First, I clean any new firearm. Give the bore a good scrub, not just a swab.
Don't leave oil in the bore and fire it. A bore should always be swabbed dry to avoid bulging the barrel. If you run an oily patch, follow it with at least two dry patches. Ditto the chamber, don't leave the chamber oily, it should be dry and clean.
For a new AR, then I fire once just to test, only one round in the mag. If OK, and bolt locked back, I load and fire two. This has nothing to do with the bore and everything to do with testing that the fire control group is working correctly. Look for doubling.
Then I load and fire three rounds. If this is OK, I then dump a full 30 round mag. Give it 10 minutes to cool, and dump another 30 round mag, using this time to rough in the sights.
Now it's "broke in".
I will then shoot 300-500 rounds before calling it a day and going home.
There I will do a normal cleaning, and put the rifle away for the next time.
If the rifle was put away with an oily bore (I do with chromemoly barrels, but not with chrome lined or stainless) then before going out for the next range session I'll run a couple of dry patches through it.
Next range session I will concentrate on fine tuning sights, group sizes. Through all of this I will watch for any malfunctions. After the first range session, there should be no more "new rifle" malfunctions. Heck, after the first two mags there shouldn't be any new rifle malfunctions.