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If you have a random orbital sander or a vibrating sander (slightly worse for this, the one that has some clips that you put a square of sand paper on) you will quickly get to a kind of nice fairly reflective mat surface finish with commonly available 2000 grit sand paper as a final step. You get a cool non uniform texture actually from the vibratory sander that I thought looked pretty neat. And it would do massive amounts to hide dirt.
Mother's aluminum polish from your will bring this up to a hazy surface with good reflective properties.
It's actually pretty nice for outdoor applications because the haze will cover up alot of dirt with little discernable difference, particularly from rain splatter. It does not look particularly reflective but you can clearly make out objects in the reflection at this point, they are just covered with haze, kind of like cataracts. 
To get a real mirror finish you will want to look on youtube to see various vendors sell aluminum polishes in a few grades that will be used with a DA (dual action) polisher with different grades of disks, I was not able to get to this point with aluminum polish that I found on the store shelves. But you still need to remove enough material to get all the deep scratches out in a timely manner, which means using sand paper if you can feel them. I would not use this stuff until you have like a 2000 grit finish from what I saw. 
Then you can cover with some kind of protectant like car wax. 
I kinda think that with surfaces that are curved or prone to dents you will be happier if you just leave it at a highly sanded polish rather then trying to make it a mirror, because all the imperfections will stand out like crazy. Highly polished stuff makes me want to hire a maid to keep it clean. 
Notice how popular the brushed nickel look is now, vs the chrome look, for things like kitchen appliances, bathroom things, modern furniture. That's the kind of look you get if you finish with 2000 grit or so on the vibratory sander. Chrome just looks trashy now.