I think the first layer (textbook meanings) can be learned just like any rote... like touch-typing. You don't (at least, I didn't) memorize where the keys are, you just learn them and then know them. Most decks would have the images as clues to each card's meaning, but even if they don't then as you set out more spreads with the deck, or do daily draws... then, eventually, every card will have come up often enough until their meaning's become familiar. In short, practice.
The other, more intuitive level of reading... well you can have a piano, even play it competently, but composing is something else. Being able to read and write notes (like knowing the textbook meanings of tarot cards) can make a good foundation for composing, but it's a gift to have one's inner music clear enough to be expressed -- or being able to pick out of the haze of intuition precisely what events, people, or concrete ideas that the cards are referring to.
This, I think is more revealed with practice, and maybe influenced by hearing other's styles, but... not learned.