Why does FM sound better than AM?

That is true for uncorrelated broad-band noise.

Correlated noise (e.g. multipath interference) and narrow-band noise (e.g. another FM transmitter) can both affect FM pretty badly.

Speaking of multipath interference, why is it that we almost never hear the effect of that? Like, aren't these waves almost constantly bouncing off of other things and being reflected? How are we not always hearing echos all the time?

The effect is easily noticeable only for high frequencies, where the wavelengths are no bigger than a few meters.

For the lower frequencies used by AM broadcasting, where the wavelengths are up to hundreds of meters or kilometers, and you use small antennas for reception, it is unlikely to have problems caused by multipath propagation (because the waves will go around obstacles instead of being reflected; only for the higher frequencies of the shortwave range you can have multipath reception of signals reflected by the ionosphere, but the objects that are around the receiver still do not cause problems).

When there is multipath propagation, you would not hear echos, because the time difference between the different paths is too small, due to the high speed of the radio waves. What you get is interference between the multiple signals, which can reduce too much the strength of the received signal. When the signal is reflected on some paths by mobile objects, or when the receiver itself is moving, the received combined signal will have an amplitude that varies in time, with intervals when the signal cannot be received (i.e. fading).

The other poster is correct but I feel there's still a simpler answer when you see the units at play. You don't constantly hear reflections as echos mostly because of the speed of light and the inverse square law.

Let's think about how far the echo has to come from to have even a half second delay. At the speed of light, that half second is 93,141mi. So it would have to reflect off something half that distance, ~46,500mi. Not a lot of good reflectors pointed at me 46,500mi away.

So then think of the inverse-square law on that. How weak of a signal is that going to be travelling those 93,141mi? Are you set up to even notice that from the noise? It's 11 billion times weaker than the original signal, assuming your reflector perfectly reflects the source signal and you're in outer space.

So obviously whatever "echo" we experience, it's not going to be something in the realm of humans directly detecting it. The shift that is possible to really mess with the signal at distance you'll actually receive reflections at are only going to shift the timing in a very small way, usually by being a slightly different phase. This means you'll get constructive and destructive interference from the same signal at slightly different phases, but not really a noticeable "echo".

Other people answered your question, but if you ever saw shadowing artifacts on an analog UHF TV station, that was probably multipath. Note that lines on an NTSC TV are scanned at over 15kHz, so this is a very small time difference.