I know wine and beer have been mentioned already but it wasn't until I seriously got into wine that I could start "naming" the flavors I taste in cigars.
I could tell you the difference between a Nicaraguan and a Dominican, but I couldn't articulate why I could tell you. Go do some wine tastings with someone who has developed their palate a bit and listen to what they say. They'll give you some BS about "tart apple, cherries, and figs on the back end" (try not to roll your eyes... they hate that) and when you taste it, you might not get two of those three, but you'll go "Oh yeah, that does sort of taste like a green apple!" That's how it starts.
Now that I'm working with guys to stand up a brewery, we test samples all the time and I'm amazed at how much better my vocabulary has gotten in expressing what I taste. It's not so much that I taste more or less than I used to, but I'm much more aware of what I'm tasting. Part of it is being mindful, and part of it is just practice. One it really starts going, it kind of snowballs. Next thing you know, you'll be talking like you're reading a halfwheel review, and you'll think to yourself "Did I just say fresh sage, night blooming jasmine, and tart orange peel behind a predominant damp hay smell? WTF?"