Before cleaning a coin, if you can make out what it is or the date, check to see if it is a rare coin or key date. If it is either, stop. Don't clean it. Next, decide whether you want to risk damaging it. If yes, continue. Any type of cleaning, especially with an abrasive, will damage the value of a coin, but with many dug coins, we just want to clean it up as much as possible to look nice.
Gentlest: Soak in distilled water or olive oil. Gently brush with a nylon brush every week or so. Heavy corrosion can takes weeks or a year or more. Not for the impatient, and also not really necessary unless you're hoping to prominently display your coin at some point. Some museums have racks of artifacts that have been soaking for years.
Quicker Methods: I find vinegar, lemon juice, etc, too be too destructive, and you have to be quite careful. Same with mechanical cleaning, like wire brushes or using a rock tumbler. You get off the gunk, but often at the expensive of damaging the coin and removing details (or in the case of one very badly corroded large cent I found, ending up with a blank metal slug.)
The best method I have found for copper/bronze coins in general is a hot peroxide bath. I use this especially on crusty Indian Heads and it works beautifully in general. It may (or may not) strip off the patina or turn your coin dark, but it usually leaves you with a gunk-free, detailed coin, with a much lower risk of damage.
My version:
1.) Heat up a couple of ounces of peroxide in the microwave for a minute (boiling)
2.) Put coin in a small container (an empty yogurt cup works well) on edge, leaning against the side; this way, the bubbles don't get trapped beneath the coin as they would if it were lying flat.
3.) Pour in the peroxide and watch it bubble. After it stops -- 3-5 minutes, maybe more -- fish the coin out.
4.) Put a bit of baking soda on the surface of the coin and gently rub with your fingers until the gunk comes off. There should be just enough liquid to make the baking soda into a paste -- add a few drops of water if that's not that case, as you don't want dry powder.
5.) Repeat with another peroxide bath if necessary.
Finally, after drying the coin, I like to put a coating on mineral oil on them, both to stop future corrosion, and to enhance detail somewhat. Put a drop on each side of the coin, work it in to the details with your fingers, and then wipe dry with a soft cotton towel. The coin should feel dry to the touch.