The problem of showing up at an open mic is to deal with the variables in the house sound system. At worst, this is one powered speaker with a couple of inputs. At best, it is a mixer with a three band EQ. You don't want to have to use the house EQ, because it takes too much time to fiddle with it. Instead, you want to have your own rig with your preferred settings which will send a signal into the house sound system with the house EQ flat.
Microphones are worthless. I played with a microphone for a while, but the irony is that it is only really satisfactory in situations where no mic is required, where the audience is quiet. If the audience is noisy, as in a restaurant or cafe, the mic picks up the crowd noise and amplifies it, and they all start talking even louder so they can hear themselves. Flamenco players can get away with this, because they mic very close and play very loudly and percussively, and if the mic picks up all the foot-stomping going on around them, that's all to the better for the show. The original poster's sad anecdote about having a mic shoved at him and not being able to hear himself is a worst case, actually. A better set-up open mic will have monitors and perhaps be indoors.
Piezo pickups are the only game in town for the classical guitar. The important detail is that piezo pickups have a very high impedance output, about 10 million ohms, and the house sound system typically wants a 600-ohm signal, stepped down by several orders of magnitude. You don't need to know what impedance is or how it works, but you need to know those numbers. Expensive pickups like those from Baggs and Fishman have an impedance transformer built in, typically based on a component called a FET. Cheap pickups like I use, made in china for 5 bucks or less (I wire three together in parallel and place them around the bridge) require an external preamp, and this preamp must have a "piezo" setting specified at 10 or 20 million ohms and an XLR output at 600 ohms. Many electronic components do not accept such high impedances.
Now, the piezo is picking up vibrations from the wood, not the air, and the quality of the signal depends entirely on the preamp or other processors used. This is why many such preamps have "tone modeling" where the sound of a known model is superimposed on your signal. These work very well.
Specifically, I am using a Boss AD-2 preamp, which is a very minimalist stomp box which contains one essential feature, a notch filter to cut the boominess in the low A string, and lacks another essential feature, which is a 3-band EQ with sweepable mids. Therefore, I run it through a Soundcraft mixer, and use a line out to go to any strange sound system. My duet partner has a Zoom A3 preamp, which although it has the user interface from Hades, has fantastic capacities for manipulating the tone. While it has no notch filter, it does have a three-band parametric EQ in software as well as a 3-band EQ with knobs. The parametric EQ can serve as a notch filter although it doesn't get in quite as tight on the frequency as it might. With the Zoom A3, you can set your sound and be absolutely confident about plugging into any house sound system and sounding at least decent. I presume that similar results can be had from the preamps from Baggs and Fishman. The Boss AD-2 needs a little help with EQ.