Before we get rolling, let's just note that Fathers and Sons is a realistic novel, through and through. Turgenev's goal is to capture the drama of a few families during a time of social upheaval in Russian history. What this means, practically, is that the story is, to say the least, symbol-light. There is no vast hidden allegory to Fathers and Sons. In other words, the majority of things in this story do not "mean" much beyond what they are.
That said, there are a couple seemingly peripheral aspects of the story that are actually much more central than the average reader might realize. The first is the role of nature. The story takes place in the Russian countryside (which we get into in "Setting"), and we often get lengthy descriptions of plants and animals in the vicinity of the characters. We'll offer just one of many examples here:
Thus Arkady. But even as he reflected spring regained its sway. All around lay a sea of golden green—everything, trees, bushes, grass, gently shone and stirred in sweeping waves under the soft warm breath of the wind; on every side larks poured out their never-ceasing trills. Plovers called as they glided above the low-lying meadows or ran noiselessly over the tufts of grass; the rooks strutted about, black and beautiful against the tender green of the low spring corn: they disappeared in the already whitening rye, and their heads only now and again peeped out from among its smoke-like waves. Arkadygazed and gazed, until his thoughts grew dim and faded away... (3.60)
What's important to notice about this passage (and other similar passages) is that nature is not just in the background. It's almost as if the natural images fold themselves into Arkady's very thought processes. What gets juxtaposed is Arkady's melancholy and the hopeful nature of the spring scene. His own brooding is essentially swept away by the "larks" and the "plovers" and the "rooks." Nature seems to both contain and direct his thoughts. Put as simply as possible: it's hard to be sad on a sunny day.
When Virginia Woolf commented on the role of nature in Turgenev's work, she pointed out that it made the human drama that much more powerful because humans were revealed to be only a "part of the whole" (quoted in Edmonds, Rosemary. "Introduction." Fathers and Sons). It is a keen observation, and to expand it, we might note that nature provides us an image of something that is more stable and lasting than the human lives at the center of his story. On a very superficial level, the debate at the center of Fathers and Sons might seem to be one between old conservatives and young liberals. Yet, when we take nature into account, we see a more powerful debate: the vanity of the young compared to those who recognize all the forces that shape one's life but lie beyond his or her control.
This is nowhere more clearly invoked than in the last lines of the novel. The narrator essentially steps into the frame of the story so as to contradict Bazarov, and his contradiction lies not only in his words, but in the images of the flowers growing on Bazarov's grave, flowers which have outlasted the young man who refused to believe in anything. The narrator says,
However passionate, sinful and rebellious the heart hidden in the tomb, the flowers growing over it peep at us serenely with their innocent eyes; they speak to us not only of eternal peace, of the vast repose of 'indifferent' nature: they tell us, too, of everlasting reconciliation and of life which has no end. (28.12)