This post contains some spoilers.
The movie opens with a shot of the sun:
This is significant, as it illustrates the hierarchies that are present in the film. As Hopper explains it later, "The sun grows the food. The ants pick the food. The grasshoppers eat the food." The whole system starts with the sun, and the movie does too.
But then a leaf falls into the water, and it's revealed that that image of the sun was just a reflection:
Distorted perception is also pretty significant in the movie - the ants are capable of more than they'd originally thought, and - more comically - the circus bugs are mistaken as warriors.
Part of the opening exposition is the ants' gathering food. At one point, Thorny says to Princess Atta, "Your Highness, I can't count when you hover like that." The first few times I watched the movie, I almost certainly didn't understand that this line is a pun (I was, like, 6 when I first saw it). The princess is hovering in the literal sense of flying, but she's also hovering in the micro-managing sense:
To contrast the regular way of gathering grain, Flik is introduced with his harvesting machine. I'd never really thought about Flik's name (because, again, when I first watched this movie, I was a stupid kid), but it blatantly indicates that he's an outcast. Flick is what you do to bugs to get them to go away. When I was a kid, I'm not sure I noticed the bug-centric nature of some of the other names either, like Manny (the praying mantis), Gypsy (the gypsy moth, although it could play into the circus connection too), Mr. Soil, Dr. Flora, and Thorny. And I certainly didn't understand the reference to P.T. Barnum in P.T. Flea's name.
Later, Flik talks with Princess Dot. He gives her a rock and tells her to pretend that it's a seed.
And then he says:
Now, now, do you see our tree? Everything that made that giant tree is already contained inside this tiny, little seed. All it needs is some time, a little bit of sunshine, and rain, and voila! ... You might not feel like you can do much now, but that's just because, well, you're not a tree yet. You just have to give yourself some time. You're still a seed.What's interesting about this is that Hopper makes the same sort of comparison later, but with the opposite intention. While Flik tells Dot that a seed can grow into a tree (that a person is capable of a lot), Hopper tells the grasshoppers to "pretend this grain is a puny little ant."
Then he throws individual grains at the grasshoppers, which don't hurt. They just bounce off:
Instead of illustrating that a person is capable of a lot, Hopper agrees with the grasshoppers' discounting of "just one ant." But then he unleashes the whole bottle of grain, and the grasshoppers are engulfed.
He makes his point: "[If] you let one ant stand up to us, then they all might stand up." So while Hopper does oppose Flik in the value of just one ant, he recognizes the power that a group would (and does) have. Ultimately both (a single person's capability and the power of a group) are part of the film's lessons.
A slightly less significant parallel is going the wrong direction. At the circus, when Manny and Gypsy are on, Manny goes the wrong way to the stage:
And then Gypsy tells him, "The stage is the other way, dear," so he changes direction:
The same thing happens later with Princess Atta and Flik. Princess Atta goes to check on the bird:
But then, realizing the bird is in the opposite direction, she crosses the screen again:
Interestingly, Manny and Gypsy and standing arm-in-arm in the background, which strengthens the parallel and seems to indicate that Flik and Atta will end up together like Manny and Gypsy are.
I also discovered some subtle references in the background of the city. First, it's Pixar's A113, along with 1195 - the release date for Toy Story (November 1995):
In a wider shot, there's a box of Darla (I think it's dish detergent) on the right:
I think this is a reference to the Darla character in Finding Nemo, a later Pixar movie, but there are five years between the two, so I'm not sure.
An-other small detail is the lighting when Dot goes to get Flik's help. At first, Flik is dismissive of the using the bird to scare the grasshoppers and thinks he himself is "a guaranteed failure." The lighting reflects this. The sky is overcast, so the colors are muted:
Then Dot gets a rock and tells Flik to "pretend it's a seed." This makes him change his mind, and to reflect this, the sun comes out, making the colors more vivid:
Flik has a literal enlightenment.
I think it's also significant that the movie ends in the spring:
Spring is usually seen as a season of renewal, and this particular spring is a harbinger of a new period in the ants' lives because they're no longer threatened by the grasshoppers.
