Start Operation Husky - the first mission of the game, after the Jump Training. Take the MP40 and the Karabiner 98 Kurtz.
Once you land, take out the MP40 and use it for the whole mission. The Italian Blackshirts have outdated weaponry, and they aren't very well trained.
If you injure one of the Blackshirts with your MP40, you should charge him and melee. The melee will give you more weapon experience than just shooting him dead.
Once you have completed the MP40, you might want to try switching to the Karabiner as well, as that will get you closer to the Upgrade 5 Weapons achievement.