Are you talking about doing the job yourself or having a professional gunsmith do it?
Either way in order to remove slight pitting, the best method is to bead or sand blast it with a blaster. Then depending on if you want a matte finish or a polished blued finish you could either leave it rough and blue it or polish it down to a smoth finish and hot blue it.
Unless it's an old traditional looking sporter, personally if I were doing this with a gun I owned I'd have all the steel and other metal parts sand blasted to a satin matte surface then have it manganese parkerized on the steel to a very dark (almost black) charcoal gray, and then any aluminum alloy parts annodized black, if any. I find polished hot blueing to be great on older traditional guns or any gun if you live in a very dry climate, but I find phosphate parkerizing to be a much better and a more utilitarian finish for a pretty inexpensive price. If you want even better corrosion resistance you can have a baked-on or heat cured epoxy matte paint applied on top of the phosphate parkerized finish. The British military has utilized this method for decades on it's arsenal rebuilt guns and have had great results with it's durability and corrosion resistance.
Personally I've always though that traditional blueing is a pretty useless and worthless finish on gun steel, just my opinion. Though I admit that there are a great many blued guns still around with their original blued finish that are still in relatively good condition that have survived use in the old west and even the trenches of WWI just fine.
The record books, that are put out by Boone & Crocket and Pope & Young and where I live is Henry Kelsey they are a great way to research a state or a province for big game. This does take some time but the knowledge it gives you, can be very valuable. Not all hunters have their trophies recorded, as where I live after the Hanson Buck became the new world record typical Whitetail, we saw a flood of hunters and now that has happened, some hunters are shying away from the record books but...