Your saw looks good to me. Ready for a sharpen. The stain on steel is very hard to remove, removing it makes no real difference to performance, so is really a cosmetic difference, is very hard to do and risks damaging the saw. Get it friction free. The Aluminium foil recommendation from CheshireChappie is also a good tip for improving the appearance.
I wrote off a saw with over aggressive polishing/etching of the blade.
I had a leak in the garage roof (boys displaced a tile while retrieving a ball) one winter, and when I got back from an overseas trip one of my old tenon saws was badly pitted. I ended up wrecking the blade trying to fully restore it. I now leave things in the state of yours, plane soles the same, rub up nice with wire wool and fine aluox paper wet.
The sand paper gets very rusty looking, but actually will keep work well if you keep it wet, you can remove a lot of rust with a small bit of paper. I recommend you either throw the used wire wool and paper away after use or keep it to one side reserved for other rust removal projects as the paper will stain wood and may contaminate other tools with rust - I keep some in a separate polybag and dispose of when really bad. But keep well away from the other abrasives.
The aluminium foil polish does an chemical and mechanical action that is very gentle. Chemically, as you rub the thin oxide on aluminum rubs off exposing reactive Aluminum metal, where this is in contact with rust it reduces the iron oxide (rust) back to iron, while the aluminium is turned into its oxide (Al2O3 fine sapphire). The aluminium foil is soft so does not scratch the steel surface. This is a surface only effect and only reacts with the rust and leaves the steel intact. You need an oil lapping agent, 3-in-one, or wd40 etc, to do this, anything oily to keep the air and moisture away from the metal surfaces, otherwise the air reacts with the aluminium before it can react with the rust reducing the effect. This is really just a cosmetic effect as the effect on the surface is really at the atomic level and has no real impact on the tools performance.
Autosol on the other hand is an abrasive and will add abrasion to the foiling action, so you will get a more aggressive finish that way. Agina its really an appearance thing as the aluox paper will have got you a friction free surface.
As others have said, oil or wax after use will preserve the surface.
Another tip, you say you are a beginner - you seem pretty experienced to me, I recommend you get going on some wood projects and then re-***** your tools, its a virtuous circle. Improve sharpening and setting technique, then do a project and then improve your tools, based on the results you are getting. I find I'm still learning after years of doing. You are going about things right in terms of learning how to properly care for and sharpen tools right from the start. You will find the limits to tools by doing projects.
If you can afford it, and the time Salaman and Paul Sellers books are a good read.
Salaman (about £25 +pp for hard back)
9780046210205 - Dictionary of Tools Used in the Woodworking and Allied Trades, 1700-1950, Used - AbeBooks . about £15 for the paper back. It was published in the 1970s so a bit dated but has good history of how things were made.
For a more up to date account, copy of Paul Sellers book its an Easter treat but can only be found new at £35 + £5pp his book and blogs on tools are a joy to listen to.
Products Archive - Rokesmith Limited I gave my 21 year old son Sellers book for xmas, he has joined a local club to learn and found it fascinating and a way to not get too many tools (we all need restraining here).
I like the way you adapted a tool to get the screws out. Sensible adaptation is a good thing as it keeps the number of tools you need down.