Other than the inside of the barrel - finish cleaning with whatever bore cleaner you use (Hoppe's #9 etc), run a dry patch, then oil a patch lightly are run that through.
For the gun, if it slides grease it, if it turns oil it. So the outside of the barrel gets greased. Grease is a thickener and oil, the thickener helps keep the oil in place.
Lube is like religion, lots swear allegiance and stick by a brand or type. You know, "I've been using XX brand for 40 years and no wear." But really it's more about keeping it cleaned and lubed with any number of lubricating products. There are lots of companies simply repackaging some bulk lubricant (even cooking oil) into tiny containers, labeling it "Gun Goop" or some such, and bumping the price times 10 or even 100 or more and raking in the money.
I use Super Lube synthetic grease by Snyco - they make a good oil too. It's translucent white, so doesn't stain like some greases with lots of color dye added, has a broad temperature range, so doesn't thicken up quickly in the cold, and is slicker than snake snot on a flat rock.
For the outside, I use Mobil Way Oil, there are many of brands of way oil, formulated for the machinery ways (tables, sliding bits). Way Oil offers excellent corrosion protection and tends to stay in place well. Besides I've got a 5 gallon pail of the stuff.
I could repackage the way oil and call it gun butter (oops, gun butter is already taken and selling for $8 for 1/4 ounce), okay I'll have to think up another name. I'll be generous though, and sell my gun snake oil for $5 an ounce - oops, that's taken too.
