Hiab
Hoppes Elite is what S&W recommends on the exterior DLC finish of the 340. It is a preservative, anticorrosive, and lubricant. You may want to use a cleaner for swabbing out the barrel. Outers Tri-Care that comes with the cleaning kits seems to be aggressive enough for the barrel and cylinder yet not harm the finish. I try to keep it in the barrel only though. I will sparingly use Breakfree on a swab or patch to run through the cylinder and barrel for cleaning.
Shooting. I would work my way up. Get some .38 spl, Winchester or Remington FMJ from WalMart and put some of it through first to get the feel.
Get some 38 spl Plus P, bulk 125 gr semi JKTD Remington boxes of 100 are around or, if you can find the Speer Gold Dot 135 in 38 spl Plus P that works, you may find it to be your favorite carry round.
Try some .357 of any kind. Winchester or Remington from Wally is fine although perhaps not the optimal carry round if you decide the .357 to your liking. Federal Premium Hydra-Shok may be around in 158 gr JHP at you LGS, I save that for my L Frame.
Again the revolver is rated for 38 spl, 38 spl plus P and .357 with no rating restriction.
You may find the later "stout". Someone said like slamming your hand with a baseball bat. Speer 125 LE, I have no experience with them, but they won't hurt your 340 so give them a shot.
On the .357s you may want to shoot 3, check the 4th and 5th round for "crimp jump". It is the bullet seperating from the casing which could cause the cylinder to "jam" or "lock-up". Shoot, the 4th and see if the 5th is still ok. This seems to be prevalent on the Sc/Ti series as opposed to the 340 with the stainless cylinder and two more ounces to "shock absorb". I have not heard this reported on a M&P 340.
This will not happen with any of the .38 or .38 plus P. I have tried many versions of those manufactured by many vendors.
Let us know what you think.