Symbolism, Theme and Subject are three different literary elements that writers use to deliver the ultimate message of the story. To understand how they work together, you must first have an understanding of the three terms on their own.
Symbolism is when an aspect of a story comes to represent more than the thing itself. Symbols are meant to be universally familiar so that the author's message is more readily understood. As a matter of fact, we are familiar with the universal in everyday life: flags, road signs and even logos. The logo of a dove immediately reminds people of peace, because doves, by nature, are peaceful. A national flag is symbolic, because it expresses something larger than itself: the culture of a country. The colors and designs on a flag are also symbolic. For example, red may be symbolic of the blood shed in defense of the country and its principles. So, symbols deliver complex ideas and concepts in a highly recognizable way.
So how does symbolism work in a story? An author might want to show the reader how an idea like "peace" or "danger" works in her story. The image of doves in the story (flying through the air or on a character's shirt, for example) might be seen during important moments calmness. In the same pretend story, the color red could symbolize "danger" (as the red muscle car of the town bully, the reddening of a character's face before conflict or the color of the sky before a major confrontation). The connection between the doves and the color read would make an impression on the reader as he reads. They remind the reader about the nature of and relation of those symbols to the characters and even other symbols in the story. If the bully in our example was afraid of doves, that might suggest that in general, dangerous people fear surrendering to peace.
Which leads us to "Subject". A subject in literature is a topic or idea the author wishes to illustrate as important to the conflicts in the story and ultimately to the message of the story. A good way to identify what subjects are prominent in a story is to answer the following question with a word or short phrase, "An idea the author wants us to think about in this story is ____." Words like "friendship", "trust" and "defying the odds" are examples of words that may come to mind. A story, however long or short, may have many subjects the author focuses on, some more than others, so it is important to think of prominent ideas that are important to the characters.
Lastly, Theme is a statement describing the message an author wants to deliver to the reader. The theme is always a complete sentence and always includes at least one Subject. As there may be multiple subjects in a story, there may be multiple themes, but typically there are fewer (but at least one) themes in a story than subjects. The way to identify a theme is to first identify at least one subject and then answer the question, "What does the author of (the Story) want to teach the reader about (the SUBJECT)?" Your answer should include the subject and a phrase that illustrates how that subject played out in the story. Here's what that process would look like after reading _Lord of the Flies_:
Question: What does William Golding want the reader to consider about SAVAGERY?
Answer: William Golding wants the reader to consider that SAVAGERY is found within all of us.
There is one more step. isolate the answer you came up with and turn it into a statement: "Savagery is found within all of us."
While many people will have slightly different ideas of the finer points of a theme or themes, most should generally understand your conclusion or at least see how you came to it if you have thought carefully about how the subject was represented in the story.
There are more advanced ways of identifying, using, and analyzing symbolism, theme and subject, so don't hesitate to ask!