Based on the picture, my best guess is that you have a sporterized M91 TS or M91/24. The barrel appears to be too short to be a M91 or M41 yet too long to be a cavalry carbine. Also, the elevation-adjustable sights preclude it from being a M38 short rifle.
The proper cartridge for your rifle is 6.5x52 Mannlicher-Carcano. Uninformed gun shop employees may often mistake the cartridge for 6.5x55 Swedish Mauser or 6.5x50 Arisaka. Ammunition is currently made by FNM, Prvi Partizan, Hornady, and Norma.
Poor accuracy is often reported with Carcanos. This is greatly attributable to the use of undersize bullets. The proper bullet diameter for the 6.5 Carcano cartridge is .268" rather than the more common .264" used in nearly every other 6.5mm cartridge. Unfortunately, the vast majority of the commercial ammo loaded for the cartridge has used the undersize .264" bullets. Only Hornady and Military Surplus ammo will be loaded with .268" bullets. Some rifles will shoot .264" bullets acceptably while others will not.
Also, the standard bullet weight for the 6.5 Carcano cartridge is 160-162grn. Most commercial loadings however, use 123-156grn bullets which will shoot low unless the sights are altered. Again, only Hornady uses 160grn bullets. Hornady's 160grn .268" bullets are also available as components for the handloader.
As far as ammo goes, avoid military surplus as it is total crap. My experience with milsurp ammo in this caliber was dirty, corrosive junk that would produce multiple hangfires and misfires per magazine. The only reason I would even consider buying milsurp ammo in this caliber would be to pull the bullets and use them as components or for the clips if the ammo came packed on them.
Hornady is probably your best bet if you want to hunt with a Carcano. Hornady is loaded with good JSP bullets and will probably cost $25-30 for a 20-round box. Norma is also quality ammo, but it uses undersize bullets and usually costs $40 per box or more
. While they also use undersize bullets, Prvi Partizan and FNM are both good options for range fun as they are much less expensive than Hornady or Norma and are good sources of boxer-primed brass if you plan to reload. Prvi or FNM will usually cost $12-16 for a 20-round box.
Graf & Sons has Hornady, Norma, and Prvi Partizan ammo in stock for this caliber as well as Hornady .268 bullets and both Norma and Graf component brass.
http://www.grafs.com/retail/