Should I change both tires when I need to change one? I know that it's a best practice in a car to change all 4 tires if you change 1, although in practice, it's only done if 3 need to be changed. Does the same thing hold true of a bike?
No.

In a car, the weight distribution is relatively even and you tend to get a relatively even wear, so that if 2 or 3 tires are obviously ready for replacement, you probably just can't tell that the others are ready. I suspect that also having 1 tire with a different amount of tread wear than the other tires on a car will lead to an unbalanced configuration that causes tires to wear out faster. (especially a left-right imbalance)

With a bicycle, the rear tire gets more weight on it than the front and will generally wear out more quickly.  By as much as 3:1. Because of this, and other reasons, some people will even use a totally different tire on front than on the back.  (could have the narrower tire on the front or on the back, depending on what you want to achieve)  On a bicycle, it's perfectly fine if one tire has worn down to being effectively smaller than the other.  You will never notice if the bike is "leaning" a millimeter or two forward or back.

The rear also tends to pick up more thorns/nails/screws/glass than the front, so there's also that reason the rear may need to be replaced sooner. The usual theory I've heard is that the front tire knocks the sharp things into a point up configuration in time for the rear tire to catch them, but I strongly suspect that it's more about weight distribution: the tire with more weight has a bigger contact patch and will push down harder on something it rolls over.

Overuse of the rear brake instead of learning to use the front brake properly can cause the rear to wear out even quicker.

You also don't want to rotate your tires between front and back.  That's something suggested with some cars, I think because you tend to wear the front differently than the back and can get a longer life that way.

However, if you're using the same type of tires on the front and the back, and it's the rear tire that's being replaced, you probably want to move the front to the back, and put the new tire on the front.

You always want your best tire on the front.  A blowout in the rear is scary and inconvenient, but a blowout on the front is very likely to lead to a crash.  You also need the traction on the front for stopping much more than you need traction on the rear for acceleration. (and because your weight shifts forward as you stop, you just can't get very much traction on the rear for stopping)

Reference: http://www.sheldonbrown.com/tire-rotation.html