Whats the best way to break in a new gun barrel. This is my first new rifle,

What kind of barrel?

Shoot it.

Some people follow very specific barrel break in.

I shoot mine.

It's on rem. 16.5 tactical barrel.

If it's chrome lined, just shoot it.

If it's not, I dunno, I don't much subscribe to break in procedures.

Shoot it, then clean when you are done.

Go to the barrel maker or gun makers website and look for break-in procedures.
The "typical" (if there is such a thing) is to fire one shot, clean the barrel - do that 5 or 10 times. Following that you shoot 5 rounds and clean the barrel - do that 5 times. The final step is to fire ten shots (slow fire, so as not to heat the barrel) and then clean it. You should be set to go.

Barrels are cheap. Shoot the crap out of it and don't clean it unless groups grow or it's going into storage.

I shoot one and clean for the first 5-6 rounds and generally by then you can see the copper fouling if any is getting less than after the first round. I then go to 3 shot groups to clean for a few groups until I notice very little copper if any. That will generally limit any fouling problems with that barrel from then on. A gunsmith I know that also shoots benchrest tells me that it just takes any fine edge off the lands or tool marks and makes cleaning easier over the life of the barrel.

I don't think it has much effect on accuracy but it does seem to reduce fouling. I have one chrome lined barrel and it showed no copper fouling after the first shot so that was it--I just kept shooting.

If it is a factory barrel, don't worry about it. Just clean it thoroughly, including de-coppering (i.e. use Sweets, Butche's bore shine, Hoppes Copper etc) after every shooting session for the first 50 rounds or so.

If it is a custom barrel, look at Krieger or Lilja's website for what they recommend.

It depends on if it fouls.
If it is smooth enough that it does not foul then break in will not accomplish much.
If it fouls then you might need to shoot short strings and clean it until it quits fouling.
Some barrels are so bad they never quit fouling.
If you are shooting an iron sight blaster it makes little difference.

Bang. Adjust sights. Bang bang bang. Adjust sights

Shoot it. Broken in. I don't hold much faith in Panther piss and frog hair break in processes