This is as loaded as a question that a person can ask. Quite a few recommend for you to shoot 5 rounds and then clean the barrel then repeat shooting and cleaning through 50 or so rounds then there are thoes that say to clean after every round intil it is broke in. Also during those 5 rounds to wait a few minutes between shots to let the barrel cool down. I have never had a problem going out and sighting in a new rifle and then cleaning it but then I can usually have my rifle sighted in with less than 10 shots. Some claim that you need to smooth out the rifling in the barrel and if you don't then accuracy will suffer. That may of been true 50 or so years ago but I believe that with the new barrels you don't have to worry about it.
There may be some difference with a bench rest shooter that needs every possible difference in accuracy but as far as I'm concerned it's just shoot clean and go as normal. I did the whole clean after every shot on a couple of varmint rifles many years ago and I never was convinced that it was time well spent.
Clean every 10-20 rounds, just so that it doesn't impair accuracy and to force you to cool it off. Barrel break in doesn't fire lap or do much of anything to smooth out rough machining. And rough machining isn't much of a concern on modern barrels. Just have fun with it.
Lots of opinions on this one. Be sure to clean the barrel before shooting. I'd then clean every 5-10 shots on the first outing and resume typical cleaning techniques after 50 shots or so. Do not allow the barrel to overheat. If it's too hot to hold, it's too hot to shoot again, IMHO.
I've seasoned and broken-in a few barrels in the past and on other more recently I have simply just fired and then cleaned em regularly after a range session. Never noticed any difference in accuracy or ease of cleaning between a seasoned barrel and one that was not broken-in. Personally I think this whole idea if breaking-in a barrel was thrown out there by the gun care industry simply to sell more cleaning products, just my opinion. Plus it's such a big waste of time cleaning after each shot for the first 10 to 20 shots, especially if the range is crowded with waiting customers. With the last two brand new rifles I bought and one that was rebarreled I didn't even bother wasting my time or my cleaning supplies trying to season or breaking-in those barrels. All three are pretty accurate guns.
Your time and effort could be much better spent bore-sighting your scope before heading to the range. On a new rifle I make the first two trips to the range simply a plinking, getting the bullets on paper, and dialing them into reasonable close where I want them. Also to get acquainted with my new gun. Nothing too serious as far as zeroing goes. After a full cleaning after the first two trips to the range I then start fine precision tuning where I want my rifle zero'd. Simple!
My rule of thumb is the same as tndeerhunter. Just never let your barrel get too hot when shooting at the range. If it's too hot to the touch, you need to slow down and take more time between shots. Too much heat with friction causes faster wear and inacuracy than anything else.
What do you do with those empty nylon feed bags or dog & cat food bags? Can you say recycle? I use them for target practice for the youngsters. Take your empty shopping bags, stuff them into an empty nylon feed bag. You can also use old clothing or rags, (but put them in the bottom so it doesnt get top heavy). When your bag is full, fold the top over, punch some holes into the top and zip tie the top shut. Paint, draw, paste, or have the kids decorate their new target. Set it out 10 or 20...