The best way to find out where to mic an instrument is with a blind test where you record it from lots of positions and then listen to figure out which sounds best to you. I did this with my mandolin and I would suggest other musicians do this too so they learn where they want to be mic'd. [3] If you're running sound for a group and have 20min to get them set up and sound checked, you can't do this.

Second best is to use your ear. Plug one ear with your finger and move the other around relative to the instrument until you hear where it sounds best. Yes, people will look at you funny, but think of it as signaling your dedication as a sound engineer. This is usually possible, but it's helpful to know where instruments tend to do best to be mic'd. This can save you time when you're in a hurry, and even when you're not it makes sense for kinds of instrument that tend to take the same mic position all the time. Let's go instrument by instrument looking at where you'd point a standard cardiod dynamic mic.

Fiddle

Most fiddles sound best when you point the mic at the bridge. This usually means putting a mic on a boom stand pointing down at the fiddle. Many fiddle players will have good posture when you're placing the mic and then slouch down 2-4 inches when you stop looking, so pay attention and adjust to where they actually have the fiddle when they're playing.

Piano

A good keyboard will usually sound much better than a real piano because there's no step where you mix the good signal from the piano with a bunch of noise from the hall. Pianos also have their strings spread out over several feet horizontally, which means that in choosing where to put the mic you're doing some amount of choosing what part of the piano's range to emphasize. This is somewhat countered on an upright because the strings cross internally. A piano will often sound the clearest if you mic it from near where the hammers hit the strings because there you pick up the sound of the attack. If you mic the underside of a grand or the back of an upright you'll lose a lot of this sound. Micing the front of an upright usually means taking off one of the front panels, which can either make the piano too loud or make it loud enough. Generally it's better to open the top panel than the bottom, because then the extra sound will be directed at the player's head instead of their legs and bandmates.

Guitar

The best place to point the mic tends to be where the body meets the neck. The bridge can also sound good, but it's hard not to get in the way there. Don't point the mic at the sound hole or you'll get a lot of the natural resonance frequencies of the body and a boomy sound. Some guitars sound really good mic'd below the player's right elbow.

Mandolin

Similar to the guitar, but much quieter. This is generally a big hassle because unless the player has learned to keep really close to the mic you probably won't be able to get them loud enough in the monitors without feedback. You can minimize feedback (for standard cardiod mics) by arranging things so that the mic can simultaneously point directly at the instrument and directly away from the monitors.

Flute, Recorder, Penny Whistle

These instruments all have many holes but the most consistent sound comes out the hole at the top by the player's mouth. This is the place the sound is generated, and the rest of the instrument is just there to adjust the length of the standing wave the player sets up. Adding a foam windscreen to your mic can help cut down on extraneous breathing noises from the player. In fact, they're generally good to have: they rarely cut out sound you want, they protect your mics some from falls, and they limit the bang when a musician forgets about their mic and stands up.

Sax, Clarinet, Bagpipes, Accordion, Concertina

These instruments change where the sound comes out based on what holes the player has opened. This is unfortunate, because their sound comes out over a wide vertical range. Luckily they're pretty loud so if you point a mic at their middle from about 8" away they should sound ok.

Trumpet, Trombone, Tuba

The sound in a brass instrument always comes out the bell. You might not need to mic them, depending on the size of the hall, but if you do they're pretty simple. Just point the mic up the bell.

Electric Instruments

An electric guitar or bass is not designed to be plugged directly into the mixer. At the very least you need a DI box, but even then you're not going to sound like the musician is used to. Much better if they bring their normal amp and play through it, and then you mic the amp.

Acoustic instruments with pickups

Not really a mic placement issue, but related. The simplest pickups are entirely passive, meaning that they use no power. These will often have a 1/4" plug. You need to use a DI box with these, and an active one will usually sound better. Others have active pickups, which need some power source to work. Most will work off phantom power and some can take a battery. Check with musicians in advance if your system doesn't provide phantom power in case they're expecting it, so they know to bring a (fresh) battery or you can borrow a separate phantom power supply. If you can't get someone's pickup to work, fall back to a mic on a stand and try to get the pickup working later.

There are a lot of people out there with more experience than I have, so if any of this seems like bad advice to you, please point it out!

[1] The farther a mic is from its target the more it will pick up the sounds of other instruments and noise from the hall. Feedback is also much more likely because the sounds from the mains and monitors will be similar in volume, from the mic's perspective, to the sounds of the instruments. Because instruments vary widely in their [natural volume] we need to be able to adjust their levels independently.

[2] Each time you double the distance from the musician to the mic you get sound that's a half as loud. So a musician going between 1/2" and 1" from the mic as they play is similar to one going between 5" and 10". If they can't hold still, you may need to pull the mic away a bit. This isn't too bad if they play a loud instrument because you'll still be able to get a good signal from farther away. It's best, though, if either they learn to stay still relative to the mic or they get a pickup or clip on.

[3] I'm thinking of proposing a session for next year's NEFFA where musicians bring their instruments and test lots of potential microphone positions to see what sounds best.