Cleaning metal

[Courk]

I'm not sure what type of metal it is, sorry. It seems mostly dusty and a bit darkened, though there might be tiny amounts of rust in nooks and crannies, I haven't looked that closely yet. What's a good way to clean it without harming it? Also, some spots have old grease on them, again, not knowing what kind of metal it is, what's a good way to remove the gunky black grease and clean the part? Does it matter what kind of grease I reapply?

[Dented Ford]

When you say "without harming it" do you mean it's some antique jewellery (you'd probably want that as "jewelry" but that'd be wrong ) item that's precious and delicate? Or some chunkier mechanical part that can bear rubbing and scraping but maybe not being dropped.
Some metals (ie silver) can be cleaned by chemical action through immersion in an aluminium (that's pronounced "al-you-mini-yum" ) pan of water with baking powder in it and heated gently. That chemically attacks the tarnish but leaves the silver intact without removing any by abrasion, as would be the case with polishing. (You do blacken the pan that way though, so you can use a wad of aluminium foil in a pan made of different metal if that's a problem). If it's a valuable silver antique article then cleaning it like this would be a bad thing anyway, because you'd probably destroy any value as an antique, as collectors value the accumulation of tarnish.
Having said all that is probably a complete waste of time anyway, as you mentioned grease and reapplying it. So... it's not going to be antique jewellery. I suppose not knowing the metal is not so much of a problem as not knowing what the article is. It is probably ferric from what you say. Depending on what it's connected to and whether there's anything but the one metal type/article included, at a guess you're not going to do much harm by soaking it in a bowl of warm water with washing up liquid in it, and rubbing gently with an old toothbrush to remove dirt and old grease. Then dry it thoroughly and repeat the brushing with WD40. That will help ease any moving part friction, protect it and sort of clean it, but it won't buff it to a shine. Does it need to be shiny?
Uh, anyway. I'm not really clued up so (especially if you can give more info about what it is) you should get a better answer from an engineering type person on here.

[Courk]

Technically it doesn't need to be cleaned at all since it will be almost entirely concealed, but I'd like it to be as clean as possible, even shiny, just so I know it's clean from that point, at least. I'm willing to bet it has decades upon decades of dirt on it, so I'm sure even just water would make a world of difference, but I want to know what's the best I can do.

For the silver trick, would it work if I got one of those disposable foil pans and used that? Is there any specific ratio of baking soda to water? I do have one necklace that I prefer to be shiny rather than black.

[Dented Ford]

As seemingly reliable a set of directions as any I've seen are here: http://www.home-jewelry-business-success-tips.com/clean-tarnished-silver-jewelry.html

[MNOWAX]

DF you're such a Brit. lol

WD-40 or gas (thats Petrol for the British here) will work for almost anything not too delicate. If it is delicate, nothing more than water and a soft bristled toothbrush should suffice.
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MNOWAX

[Courk]

I was just thinking of the interior, but I just realized that pieces on the exterior that I assumed were to be painted actually need to be super shiny. The exterior is actually one big hunk of metal that needs the painted areas to be repainted, so whatever you can recommend to get every speck of dirt off of it without losing the details in the metal.

[worm]

do you know what the purpose of the piece is?

how about a picture, so we can get a look at the grease and maybe tell something about the type of metal?

[Dented Ford]

Probably a domestic wire wool scouring pad (pan scourer) to begin, if you're not too heavy handed, especially to get rid of any old paint residue. Then go to very fine glasspaper and gently smooth. Then go to a regular metal polish and cloth, especially a chrome polish type.

[MNOWAX]

DF has it right there, i think.
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MNOWAX

[Dragon Phoenix]

Ultrasound.
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[Dented Ford]

[MatthewV]

Lemon juice or ketchup can make metal shine with little or no effort.
I would use a light oil instead of WD-40. Solvents are icky. Oil is needed mostly in places that should be moving.

[Courk]