Mosin actions are held in place by two machine screws; both should be tight. When both are snugged up, the magazine/trigger guard is snugged tightly against the receiver. Ideally, the gun stock should be firmly sandwiched between those two items, with the recoil lug snug against the cross bolt in the stock.
Don't worry too much about the bolt being loose when the rifle is cocked. They all wiggle a bit, but as long as the trigger pull is tolerable you should be OK.
Other tips, in no particular order:
* Clean the inside of the chamber thoroughly with a shotgun bore brush. Lacquer residue from lacquered casings gets stuck in there, causing the dreaded Sticky Bolt. The previous users of your rifle may have been meticulous about cleaning the bore, but the chamber is often another story entirely.
* The corrosive salts left over from shooting old corrosive-primed ammo are water soluble. Ammonia is not strictly necessary, but it won't hurt. Since I shoot black powder in other guns, I just use Birchwood-Casey No. 77. Plain old water works too--just leaves more of the powder and copper behind for your petroleum-based solvent to deal with. In the field, a couple patches saturated with your own spit is better than nothing.
* Several ways to adjust the trigger: (1) Polish the contact area between the trigger and the spring/sear gently by hand. This will make the pull less gritty but not lighten it. (2) CAREFULLY stone the sear and corresponding spot on the bolt. This will FUBAR your trigger if you do it wrong, but if the trigger is still gritty after the first step, you can either live with the grit or take matters into your own hands. (3) Fit a shim under the spring/sear where it is fastened to the bottom of the receiver. This will lighten the pull and reduce the amount by which the sear engages when cocked. If you overdo it, the rifle may fire when dropped, so be careful. It may also free up a bit of pre-travel in the trigger before it encounters spring tension. The Finns often installed a light coil spring directly on the trigger pivot to take up the pre-travel.
