I think the answer for most people is: we don't. At least not in any structured, scientific manner.
Personally, I eat very intentionally when I am trying to learn more about a dish and its ingredients. What I mean by that is I try to break down the dish into its component parts and really think about what flavors I am detecting.
I've listed some guidelines below. Consider how you can incorporate these into your everyday life. While some kind of focused training regimen could work (it sounds like a fun experiment), these are things you can do every single day to improve your knowledge of flavor.
- Eat lots of good food and be mindful of flavors as you eat.
- Taste often while cooking (including your raw ingredients) to learn about how flavors change, interact, and layer.
- Learn about classic flavor combinations from various cuisines around the world and try to decide what makes them work and how you can twist them.
- Slow down.
- Change things up and try new flavors often. You can't learn how an ingredient tastes if you don't eat it.
- Be fearless. People are often hobbled by preconceived notions. "Oh I don't like asparagus." Well that's all fine and good but why don't you like asparagus? For almost any person can find ways to appreciate almost any food given a positive attitude and some experimentation.
- Your palate is just as much about your nose as it is your mouth (perhaps more-so), so lots of smelling is in order.