What is the easiest way for me to measure the amount of "juice" that's left in an alkaline battery?
If we talk about small AA, AAA and coin batteries (both 1.5V and lithium 3V), then the simplest estimation can be made using a voltmeter. The voltage goes down (declines) over time and usage.

Note that each particular device needs a different minimum voltage to operate normally, so for example a battery from one device which seems dead can be put somewhere else and operate normally (at least for a little while). For example my digital weigh is quite demanding - it ceases operation already when the voltage drops to 2.88 V, and those lithium batteries are still well usable in bicycle computer or somewhere else.

Update: Note that alkaline batteries have got some kind of self-healing effect. When they are not used for a while, their voltage goes a bit higher. This can mislead your voltage measurements. So you should put them back to some load for a few seconds and then you can see their real health. Also note that totally dead batteries don't need this; they show voltage near to zero even without any load.