Personally I have had good luck by starting with wet/dry paper around 400 grit then going to lighter and lighter grit, up to 1000, then hitting it with my Dremel and buffing compound. Time and patience is the key; don't be in a hurry.
If it is the standard Ruger brushed stainless finish a 'Scotch-Brite' pad works perfect. A slightly used green one seems to work best for me. Be sure to brush with the finish grain.
I may be preaching to the choir, but do not use an iron wire steel brush or steel wool on stainless steel. Iron particles will embed into the SS and you will get surface rust.
First I'd try polishing paste and see if that works before I do sandpaper or scotchbrite. If the paste doesn't work, I'd try 4X steel wool (very extra fine). Sandpaper, 1000-2000 grit would be last to a mirror polish.
VERY lightly use the grey scotch-brite pad. DO NOT be in a hurry. Wipe in ONE direction only, never back and forth. Wipe in the direction of the brushed finish. Use good lighting. Stop and inspect often. In this manner you will do a perfect job, every time.
If you want to polish the barrel and it happens to be round, get someone to help you w/ 1000 and finer emery and give it the shoepolish treatment. Two hands on emery and pull on ends w/ barrel in the middle.