Instructions
Pick a good one. Mosin Nagant barrels can vary greatly. Make sure yours has strong rifling and a good crown (the business end of the barrel). Don't be afraid of a counter bored barrel either. This was generally done to improve accuracy.
Clean it thoroughly. Heat and any solvent that will work on oil based products will help you remove the packing grease, cosmoline is the actual name, from your new rifle. Of course, ensure you follow the manufacturers recommendations when using solvents, as many are noxious and flammable. When the grease is all gone, clean the rifle like you normally would. Be prepared. These guys take some time to clean.
Try different kinds of ammo. Mosin Nagants shoot 7.62x54r ammunition. This ammunition is available in a number of variations. Some MN's will shoot some ammo better than others. If you are using milsurp ammo, buy several types and then stock up on the ones that your rifle shoots best.
Cork the barrel. Try using a small shim or two of cork gasket material between the front of the stock and the barrel on the under side of the barrel. This sometimes helps relieve pressure caused by warping in the wood over many decades. In extreme cases, you might consider buying an after market Mosin Nagant stock. These are readily available and can improve accuracy if bedded correctly. If you decide to go this route, please keep your original stock as a sign of respect for the history of your rifle.
Check your action screws. Almost all MN's will have loose action screws when you first get them. Make sure you tighten them up. If an action screw is missing or broken, there are a number of websites that sell firearm parts. You can also find them on popular auction websites. You don't have to be superman when tightening your action screws, but make sure they are snug.
Shim the pillars. A few thicknesses of business card under the action pillars can sometimes lift the rifle slightly in the stock, relieving pressure.
Consider after-market gun sights. Many people have trouble with the stock iron sights on these guns. Consider an after market peep sight or a long eye relief rifle scope with a quality scope mount that replaces the rear sight. You will spend a little to get quality here, but it's worth it for a scope mount that holds zero.