If the scratches that you're trying to remove are deep or pronounced enough you may need to start with a bit more abrasive grade of sandpaper or steel wool to remove them. From there it's just a matter of using progressively higher grits until you have a look that you like. I've put a high polish on quite a few objects that I own and in my experience the brushed look happens at around 1000 grit, maybe 1500 if you want the scratch pattern to be less pronounced. A poster above suggested green scotchbrite and turning the mod in the pad. It's a good idea no matter what you use to apply the finish because it allows the scratch pattern to run evenly in the same direction and that really the most important part of the finish. It won't look right if the pattern isn't even. Also, sometimes the grit of the abrasive used isn't as important as applying equal pressure which is why turning the mod in an abrasive pad works so well. It allows equal pressure on all parts of the mod. Sorry to be so long winded but I've done this kind of work a lot because I'm almost never satisfied with the factory finish on items of stainless steel that I own. The good news is that when polishing, blasting or brushing stainless steel nothing is you do has to be permanent. If you don't like your work just go down a few grades an start over. Post up some pictures when you arrive at a finish you like.
Chuck it up in a lathe if you want it perfectly symmetrical for a brushed finish. There's also bead blasting for a matte peened look, various grades of aluminum oxide blasting, and various coatings that can be taken advantage of. I enjoy metal work and various finishing techniques.