The file is to take out the tool marks, not always neccessary depending on how nice a finish you got off the mill. 150  or 180 paper will take out the file marks, 220 will take out the 180, 320 takes out the 220 etc. 
Keep your file clean or it will put BIG tracks in from the metal chips stuck in the teeth (sometimes called pinning). To clean, smash the open end of an empty brass cartridge in your vise and use the resulting flat end to push parallel with the teeth and they will come clean easily.
When polishing, change angle of strokes between grits so say you are sanding the 180 scratches out with 220, then you can see when all the 180 scratches are gone. If you don't do this and sand all parallel with the barrel you finally see those remaning 180 scratches show up about at 320. Nothing to do then but go back to 220 and re-do.
Use a file or plastic bar or hardwood board to back up the paper, don't use something soft or the flats will not look good after they are blued.
How fine of grit depends on what look you want and how you are going to blue.  Hot blue gets shinier and blacker to my eye the finer you go and doesn't look right to my eye on an old time gun above 320 grit. Rust blue will most likely knock a high polish back to the 320 or coarser depending on how agressively you rust it.
You can dust a 320 finish with very fine glass beading or other abrasive media, hot blue and get a "matte" effect somewhat (but not exactly) similar to rust blue effect.
A little ibuprofen might come in handy before you are all done.
kootnre

Kootne pretty much covered it.  I do much the same.  I use a block of wood 1x2x6 inches, smooth planed oak or maple.  Wrap the paper around it, and use it like a file, just like you are draw filing.  I don't hesitate to use 80 grit paper following filing.  When the file marks disappear, go to 100 grit, 120, 150,180,220,320,400.  Doing a thorough job with coarser grits saves time over trying to take the same amount of material away with finer grits.  I'd much rather sand 3 times with 80,100,120 than try to clean up file marks with a single episode of 120.

Sandpaper gets dull, although it seems most people don't recognize that and sand with a sheet until it wears through.  I buy my sheets by 100 packs, and the price is usually below 50 cents for cloth backed aluminum oxide, and a bit more for the finer grits in silicon carbide wet/dry.  Compared to what my time is worth, paper is cheap.  I work for about 2 minutes with one side of the wrapped board, then rotate and use the opposite side.  Then, re-wrap the paper to offset the side seams onto the faces, and use up those two sides.  Then rewrap the used sides onto the board first, and use up the unused half of the paper.  I cut the width of the sanding sheet so it makes 2 complete wraps on the board, so that this method works.  In the other dimension, cut the paper slightly shorter than the board so there is none unsupported.

For the coarse grits, it takes 2 pieces of paper wrapped on the board to sand a 32" octagon barrel.  For the finer grits, usually one piece is sufficient.   

Doing the math:  that's 30 minutes per 80,100,120,150grit, 15 or 20 minutes for 180,220,320,400.

4 hours of hard work for me to polish a 32 inch octagon, after its draw filed.

If it's smooth enough to not need draw filing, add another 30 minutes for the initial 80 grit to take out the tool marks.  Otherwise, add 1 to 2 hours for draw filing.   

My hands can't take that much at once, so that's spread out over 3 or 4 days.

dave

I wondered about factory time spent on a barrel too.  I'm already about 6 hours into this and still not sure how it's going to look.  I don't quite know where to stop.   I did figure out what Kootne said about the  180 marks re-appearing, tho.   I am hopeful that it will look like I know what I'm doing.    Probably get round  or 1/2 n 1/2 barrels in the future.

-06

The old time factory guys didn't finish flats the way we are talking. They used a machine (Planer) with a reciprocating tool post holding leather or cotton duck impregnated with oil and emory. This was clamped in a spring loaded arrangement so 2 opposite flats were done at the same time. You can bet nobody stood around doing barrels the way we do. Big important job in those days was called a "Time & Motion" man. His job was to watch other people work at bench or machine and work up processes to eliminate all ineffecient methods. I believe round barrels were also done this way by some manufacurers. Look at old un-redone barrels under the forends with a loupe and you can see the tracks are lengthways.
kootne