To clean up saws generally first I try to remember to take pictures (because to often I just get excited and dive in and then never have before pictures). Then I take the saw apart, essentially just remove the handle.
Once I have the saw disassembled I normally start with cleaning up the blade. I use wet/dry sandpaper to clean the blade. If there is a lot of rust on the blade I will use 400 grit but generally I try to primarily use 2000. I also use a block of wood to sand with. If you only use fingers or something spongy to sand with that it is much easier to remove or sand way any etching or stamp on the saw blade. The hard wooden block makes sure that the sandpaper stays above the etching, unless the etching is already pretty much just worn off. While sanding I also try to keep way from the 1/8″ by the teeth to help make sure I don’t sand down any kerf that is already present on the saw.
Some people also put gun-bluing on the blade to help show the etching and to give it strength and protection. I have not tried this yet.
Next I like to sand down the handles. I know that some people just prefer to clean the handle or try to find ways to clean the handle without removing the original (or current). So far I have really enjoyed sanding the current finish off the handles to apply a linseed oil or some other type of rub-in oil. I do not try to remove every stain or blemish from the handle just try to get it back down to the original wood to refinish. I think that the dents and dings of the handles give it some charm and if I just wanted the handle to look brand new it would likely be easier just to make a brand new one.
Lastly, I clean up the medallions and buttons/pins some. I use a light hand wire brush just to take off any dirt. I don’t necessarily try to polish them to a shine but do try to get the dirt, grime, or other such stuff off of them.
If I’m going to tune the saw myself I will do it now before putting the handle back on. Then I put the handle back on and find some scrape to try the saw on. After the I try to remember to take the after pictures.
Let me know if you are wanting any more information or clarification on a step. That’s how I refurbish my saws.