Some of the things an "experienced" eye will see, though not all on any one coin, usually:
1. Fine to deep scratches running parallel to one another across the high points (devices) of the coin and/or larger areas of the flats (fields.) Or, alternately, swirls of scratches, again running parallel to one another, scream that the coin has been cleaned. (The normal scratches, dents and dings on the surface of a circulated coin are random in their arrangement and depth.)
2. Shiny or lighter colored fields than the denticles, stars, dates, mint marks or other devices. Usually you would expect the high points to be shiny/lighter colored, but cleaned coins tend to be just the opposite, depending on how long ago the cleaning took place.
3. Dirt and crud built up around the devices while the fields and other easily cleaned surfaces look, well, clean.
4. On copper...any pinkish tinge or unusual toning, especially of older coins (say from the 1800's or early 1900's) is reason for concern.
5. On silver, if there is no tarnish or toning at all on otherwise obviously circulated coins, the coins have been cleaned. Such coins appear very "flat" to the eye. Even some AU and BU coins have been cleaned, and may, upon casual inspection, seem fine, but you will often see less original mint luster than you would otherwise expect, even if the coin is "shiny." This is, for me, the hardest to spot and takes a truly practiced eye. (It may be useful to visit your local dealer if you have a chance and look at as much of his/her inventory as you can without being a pest, just to get used to studying coins in various states. This is especially useful if you are intending to collect a particular series.)
6. On silver, be sure to look closely at heavily toned coins, because the toning is often created artifically and may well cover evidence of cleaning.
Grab some circulated coins from your pocket change that have no numismatic value. Clean 'em. Use soapy water or Brasso, or whatever, and study the results. Leave them open to the air for days/weeks/months and watch what might happen to the coins as surface oxidation results in toning or at least darkening of the surfaces.
In my opinion, you will have a hard time finding 18th and 19th century
US coins that haven't been cleaned at some time in the past. It was common practice and actually "preferred" by a lot of collectors.
Sometimes you can get a very good deal on a cleaned coin that is otherwise nice. Carry it in your pocket for a few days or weeks, and it can begin to get some character back, and the cleaning will be less noticeable.
Finally, remember that even experts will sometimes disagree on the state of a coin...cleaned or not cleaned. It is up to you to decide; don't believe everything you hear, see or read about a particular coin. Study the coin prior to purchase.
Best wishes.