I have two Mosin-Nagants, an M44 and a 91/30. They were my first guns ever, and I still shoot them. I prefer the 91/30 to the M44 personally, as it is better balanced (after all the M44 is really just a 91/30 cut down for urban combat) and is definitely the more accurate of the two. YMMV, and I think that the bore on my 91/30 is better than my M44, also the improved sight radius is a probably factor. I am proficient enough with my 91/30 to bust clays at ~225yds (set on the berm behind the 200yd target boards), and though it's not a tack-driver, it is accurate enough to help me continue to improve my shooting technique. My 91/30 is still my favorite range gun, though the 18.5" 870 shotty is quickly catching up to it - two completely different kinds of shooting, though.
I have a small stash of Hungarian yellow silver tip ammo (steel core heavy ball - 182grn) and Czech silver tip (steel core light ball - 147grn) that perform well. You can find the stuff for decent prices online (I use Ammunition To Go in Brenham and buy it in person). If you're looking for a very good accuracy round, I'd recommend the S&B 147grn rounds. At the price the stuff runs, it's not range ammo, but would be great if you want to go hunting or some fun times seeing what your gun is capable of at the range.
Here I am at the range with her, bayonet and all:
Great guns, the Mosin-Nagants.