Steps to take:
1. Make sure action screws are tight. If they're not, everything moves with every shot.
2. Clean the bejeezus out of it. My M44 was so dirty that I still get more definition in the rifling every time I clean it. Even so, it was a 3 MOA gun with Czech Silvertip, over iron sights, with my eyes.
3. Shoot good ammo. The cheap surplus stuff is so inconsistent as to be noticeable to the shoulder. Bang...bang...bang...bang...BANG!!!
I haven't shot the Winchester or Hot Shot (re-labelled Prvi) for groups yet, but a back to back comparison shooting at a steel plate at around 200 yards showed it much easier to get the hits with the Winchester than with surplus. This while sitting on a big rock, not off a bench. My first batch of handloads grouped better than the surplus stuff, when I could figure out where it was landing (it by no means shot to the sights).
4. Put a slip-on recoil pad on it. The Butler Creek small pad fits the Mosin stock pretty well, and both tames the recoil and adds some much-needed length-of-pull. They're about 15 bucks at your LGS.
5. Shoot it a lot. This is where cheap surplus ammo really has it's place. Learning the rifle make a big difference in group size, and a Mosin is about as different a rifle as most of us are likely to shoot. When I first got my K31, I actually shot better groups with my M44. Why? Because I knew the rifle. Now, of course, the Swiss rifle shoots rings around the Mosin, as it should.
6. Buy a bunch of cheap ammo (I like the Czech silver-tip, although Hungarian yellow-tip has a pretty impressive flash), go out to someplace on BLM land where you can shoot at whatever you see, and start blasting rocks, Coke cans, tree stumps, whatever. Do not care about groups, MOA, measurements, whatever. Pick a target, verify that it's safe, shoot at it, look for the dust cloud, adjust sight picture, repeat until out of ammo.
--Shannon