Here we have another controversial "coin cleaning" thread. Put your flameproof jacket on, Nic; coin cleaning debates can get pretty heated on this forum!

Many if not most collectors regard the removal of toning as "cleaning" and therefore something that should not be done; I am one of them. Others (including more than a few dealers) believe "shiny is best" and remove toning at whim.
One thing that should be clear from the start: you will not be able to make a circulated, toned coin "shine like new". Mint lustre is created when the coin is struck, and the only way to put it back once it's gone is to melt it down and make a new coin out of it. Remove the toning from a coin that's lost its lustre through circulation or corrosion and it will look dull, flat and "cleaned".
You certainly should not remove toning by mechanical means, such as a wire brush, toothbrush, polishing cloth, gemstone polisher or anything similar. This will almost always make the coin look worse and, in the rare instances when the coin looks better, the coin is deemed to be "whizzed" - a practice which is potentially both deceptive and unethical, since it attempts to artificially recreate the lost lustre.
Chemical removal of toning is the usual course, and probably the least harmful. It still does harm (the damage can be seen under a microscope) and should not be undertaken lightly.
The chemical you use depends on which metal the coin is made of; despite what some of the proprietary coin cleaning formulas might claim, there is no "one chemical cleans all coins" substance... unless you count extremely strong acids such as aqua regia (which you
definitely don't want to be using on your coins). Whatever chemical you use, always rinse it off with plenty of water once you're finished cleaning. And NEVER dip coins made of one metal into a solution you've just used to dip coins made of another metal; you could end up depositing dissolved metal from the first coin onto the second coin.
Gold coins don't need cleaning. If your gold coins are tarnished and need cleaning, then I hate to break it to you, but they're not real gold...

Silver toning, or tarnish, is best removed with an acid-thiourea dip. There are many that say a quick dip is harmless. I disagree; at best, a quick dip is "mostly harmless". At worst, it will make a coin dull and lifeless.
You can also use an autoelectrolytic cleaning method using a piece of aluminium and some warm water. Dissolving baking soda in the water can help if the tarnish is thick. On a related note, DO NOT store a heavily toned silver coin in a loose bag, bucket or jar with a pile of nice shiny aluminium coins for months; the moisture from the air is enough to complete the circuit and you will end up with a gooey mess, in the middle of which will be a mostly-cleaned silver coin surrounded by a pile of ruined aluminium coins. Trust Sap on this.

Nickel and stainless steel coins shouldn't need cleaning since they don't tend to form a "toning" layer; they're more likely to corrode than tone. Cupronickel coins do tone, but beware of using acids - some of them can strip the nickel from the surface, leaving the coin a coppery colour.
Copper, bronze and brass coins: commercial brass polishers typically contain ammonia and an abrasive powder; not something you really want to be putting on coins. A mixture of citric acid (lemon juice) and salt is a common "coin cleaning mixture" you can find on the web and in school science textbooks, but most acidic copper cleaners make a bronze coin turn an ugly pink colour and the resultant surface often becomes "activated" and quickly retones. When cleaning such coins, especially pure copper coins, rinse them only in distilled or deionized water; ordinary tap water will leave chemicals behind after it dries that causes verdigris formation.
To conclude, and in case I haven't been clear, most collectors by far prefer that toning is never removed.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis