Stainless barrels can be very forgiving, but once they stain, they can be a devil to take back to a nice shine. J-B Bore Paste is what I use on a stainless barrel. I have also used Brasso cleaner, but like the bore paste better.

One other thing I forgot to mention.. with the trigger you do not need to do this every time you shoot. Normally a simple wipe of the fowling along the top of the trigger area will be all that is needed. I clean my triggers about every third or fourth trip to the range. But do make sure you do the maintance or one day you will discover that the bolt is locked up and the rifle will not cock the hammer.

Also when you are done cleaning the rifle, and putting all the parts back together, the breech plug goes in first, then the trigger is put back on the barrel before you slip the bolt assembly back into the rifle. Other wise it does not seem to catch and work properly...

Keep us posted how well the rifle shoots... Good luck.