One place to start is with a symbol dictionary. This will tell you what symbols have come to mean, mostly in Western culture. A symbol dictionary represents a kind of mutual agreement perspective (some say based on human instinctual 'archetypes') found in art and literature so that we can expect others will probably recognize the same symbol in a similar way.
Some people or groups develop highly personalized meanings for symbols that seem to work just as well as the broader cultural understandings.
Despite certain conventions, not all tea cup readers will perceive the same symbols in a cup and even if they do, they might interpret them differently. So there is an art to the practice.
I could never see anything in tea leaves on the few occasions (over many years) when I tried. Then I took a workshop taught by Shaheen Miro at the Denver TarotCon and suddenly I saw the tea leaf symbols very clearly and immediately knew what they meant. It was like a revelation. The key after that is practice, practice, practice!
Some people see symbols all around them - naturally and instinctively, some develop the 'sight', and some can point to meanings in a symbol dictionary and piece together an interpretation like mechanically learning a language. Anyone who is exceptionally good at any of these ways (and others I haven't mentioned) can become a good symbol reader.
Although not specifically on divination, some Jungian works will serve you in good stead for learning about interpreting symbols - especially Marie-Louise von Franz - for instance,
The Interpretation of Fairy Tales.
http://www.amazon.com/Interpretation-Fairy-Tales-Marie-Louise-Franz/dp/0877735263/
Another good book that both talks about symbolic interpretation and includes an excellent symbol dictionary is:
Ariadne's Clue: A Guide to the Symbols of Humankind by Anthony Stevens.
http://www.amazon.com/Ariadnes-Clue-Humankind-Princeton-Bollingen/dp/0691086613/
You might also find the following book helpful:
Pictures from the Heart: A Tarot Dictionary by Sandra A. Thomson.
http://www.amazon.com/Pictures-Heart-A-Tarot-Dictionary/dp/0312291280
A good psychic development class can help with the "seeing" part of it - which is harder to do on your own if it doesn't come naturally. Anyone can learn and learning it in a group usually augments the learning tremendously.