If I need to cool something quickly or keep something at a lower temperature, do I put it in the top or bottom shelf of the fridge? Which part of the fridge is the coolest?
Most refrigerators have the cold air outlet on the top, that way the cold air sinks down naturally to the bottom. Air from the bottom is then circulated through the heat exchanger to make it colder and then goes back in the air outlet at the top. So the very coldest place is right in front of the cold air inlet in the top. That's generally below freezing though, so you may get things colder than you want.
Other than that the coldest place tends to be the bottom shelf. You could also cool it down in the freezer for awhile and then transfer it to the refrigerator.
In general, unless your refrigerator manual says otherwise, the lower parts of the compartment will be the coldest. This is in line with simple physics, as cold air is denser, and tends to sink. You also don't want to store foods that need the coldest spot on the shelves built into the door.
In general, you may wish to take more proactive measures to rapidly cool food.
See also What are some techniques to cool down a dish so that we can store it in the refrigerator safely? for specific advice on this.
@GdD is right, the coldest part is usually the back of the top shelf as the coldest air blows directly on it.
The counter intuitive point is: the cold air won't survive long enough to make it to the bottom unless all the shelves and food in its path are at least as cold.
Otherwise, the top of the fridge would remain warmer and all the cold would settle in the bottom. The fridge is not an ideal system, for the chimney effect of the cold air to work fast enough to make the bottom coldest, the temperature gradient in fridge would need to be a lot higher which would mean a badly designed fridge.
Fridges usually blow coldest AIR at the top and it slowly works its way down and form a gradient. Only after the top shelf has had a chance to cool down to the incoming cold air temperature do the lower shelves stand a chance of getting cold.
This is even true with those little bar/office fridges with freezer inside at the top. If you find the milk slightly frozen, move it DOWN a shelf and it stops from partial freezing.
Often the back of the fridge is the coldest, and the bottom is colder than the top if the fridge isn't too full. The door isn't quite as well insulated, and when you open it the front gets warmed up a little more than the back. It's reasonably common to find things frozen at the back of the fridge, especially on the bottom shelf, while the rest of the fridge is at the desired slightly above freezing temperature. If the fridge is more full, the top may be colder because all the food is blocking the cold air from actually getting to the bottom.
If you have drawers at the bottom, though, they're likely to be warmer in practice, because they get periodically warmed when you open them, and are slower to cool back down since they're mostly cut off from direct air flow.
Beyond that, wherever the cold air is being pumped into the fridge is the coldest. The vents should be visible, and you can feel for cold air flowing while the fridge is running. But the vents are usually at the top, not right next to a shelf, so that the air can flow around instead of freezing whatever's right in front of it. Depending on your fridge, it may or may not be possible to stick something right in front of the vent.