First off - the supplied cleaning kit is cr@p.
If you get stuck out in the woods and have shot corrosive ammo, and are now looking at 5 weeks without a chance to clean it, then ok, use the milsurp kit. Otherwise, don't bother. You'll most likely damage your gun using that kit, either through damaging the rifling and muzzle, or through not being able to clean the gun properly and having the bore rust and pit.
Using a good commercial kit, I do the following with my nagant when I shoot corrosive.
1) remove the wood stock.
2) brush the bore quickly, and rinse the brush in hot water. Usually this frees up a lot of the crud in the bore.
3)Get a plastic funnel, put it in the bore, pour boiling water down the barrel. Pour as much as you like - the more water the better. Wear oven mits!!!! The barrel gets very very hot (boiling).
4) scrub the bore with both a brass brush, and a cleaning jag with cloth. On a newly bought mosin, it could take a long long time to get the barrel clean. Having several brass brushes and lots of paper towels and small pieces of cloth to swab the barrel out with will help a lot.
Remember, with a brush, push it ALL the way through, then reverse direction when it is out of the bore. If you push the brush half way, stop, then pull it back it can jam up in there, and possibly damage the bore... or at least just get really really stuck.
5) If you want, scrub the bolt assy and reciever with hot water in case any corrossive salts have gotten on them.
6) re-oil the bolt, trigger assy, bore, and all exposed metal surfaces to avoid rusting.
DONE!
I may be overly cautious, but I like cleaning my guns, so I don't mind.