When is it better for me to shut down my Kindle Fire vs. putting it to sleep? Does putting it in sleep mode drain the battery?
In most cases, you will not need to shut down your Kindle Fire. The Fire uses a lithium-ion battery, which means you should be connecting it to your charger quite often. (Shallow discharges and frequent recharges are better for your battery in the long run than deep discharges.)
The only time you would gain anything from shutting down your Fire would be if you were sure you would not be using the device for an extended period of time (more than a couple days). In that case, the recommended charge for storing a lithium-ion battery is around 40%* - so you should discharge the device until it comes close to that amount and then shut it down.
The Kindles are designed to be turned off only rarely (such as for lengthy periods of non-use), and to rely on the sleep function instead. Nathan's answer covers that admirably.
However, you may wish to turn off the wireless connection when you put the device to sleep if you think you might not pick it up for a few hours. That will considerably slow its power use.