1. The starting point is face value. If your note, or bill, is less than 50 years old, it is probably worth only its face value. Likewise, if your coin is less than 50 years old and was minted for circulation -- an ordinary quarter, for example -- it is probably worth only its face value. I mention face value because it does establish a floor for future value when you buy a coin or note; the U.S. government has never pulled the plug on old coinage and currency. The other two valuations I'll describe go up and down with the market.
2. The underlying paper in a note has no value except to a counterfeiter, but most coins are worth at least their value in metal. If the coin is composed of a precious metal such as silver or gold, this metallic value will almost certainly exceed the face value. Coins of precious metal that were never intended for circulation, such as the Maple Leaf from Canada, are called bullion coins. Platinum and palladium are used occasionally in bullion coins. At one time, coins minted for circulation also contained gold or silver.
You may not know, however, some of the complexities of this valuation. Pure precious metals are so soft that they will not withstanding frequent handling. In most cases a less expensive metal like copper is alloyed into the coin to increase its hardness. Sometimes the metallic values of older coins vary as a consequence, although in recent years the U.S. Mint has put the same amount of gold into its American Eagles (which are alloyed) and its American Buffaloes (which are not). The Eagles are simply a little heavier on a scale. When you hear the weight of a bullion coin in ounces, be mindful that the reference is to the troy ounce not the common ounce. Age of the coin is not a factor in metallic value, aside from the history of alloying, and up to a point neither is condition of the coin.
Cherry-pickers have long since captured nearly all circulating coinage that has significant metallic value. Your chances of finding a silver coin in pocket change are microscopic. Ordinary pennies and nickels these days have a metallic value above their face value, but no one cares because it's too onerous to bother with them. Besides, in 2006 it became illegal to melt pennies and nickels to reclaim their metallic value or to export them for that purpose.
Caveat: as gold speculators have seen recently, prices for precious metals fluctuate.
3. The final approach to valuation for both coins and currency is as a collectible. Many factors come into play: rarity, eye appeal, age, condition, the site of the mint or printing plant, authentication as genuine, etc. Two examples: if you have a $10,000 note -- the largest ever placed into circulation -- it is almost certainly worth at least $25,000. If you carry a 1909-S VDB penny in your pocket, for heaven's sake put it into a safety deposit box; even a worn specimen is probably worth $1000. Proof coins and commemorative coins are primarily collectibles. The true value of a collectible is whatever a buyer is willing to pay for it at a given time. Hobby publications provide approximate values based on reported transactions for the most frequently traded collectibles. The really good stuff goes to auction. Caveats: subjectivity and desire often enter into these transactions, and overall prices for collectibles can fall.
In short, your coin or note is worth the highest of these three valuations. What should you watch for in everyday life? I take a second look at notes whose seals are not green, notes larger than $100, notes or coins older than 50 years, and notes or coins that have an obvious production error (such as a quarter on one side and a nickel on the other). All else, I ignore.