Newton's third law of motion states that every action has an equal and opposite reaction. That is the reason we do not sink into the earth, because when our weight exerts a force on the earth it also exerts an equal and opposite force on us.
But when we stand on quicksand or on fluids we can sink in. How is this possible? Does it not exert an equal and opposite force on us? Or are Newton's laws different in the case of fluids and substances of low densities?
This is a common confusion when people first learn Newton's Third Law. They get the idea that it implies motion can never begin.
The (wrong) argument is that, since the Earth exerts a force $F$ on you by gravity, the sand must exert an equal and opposite force $N = -F$ on you. Then the total force on you is $N + F = 0$, so you can't fall. Of course this argument can't be right, because it either means that nothing can ever start moving, or that Newton's laws don't apply to sand, as you propose, and neither make sense.
What Newton's Third Law really says here is
It gives no relationship at all between $F$ and $N$, so you can indeed begin to fall.
Read the law well!
If body A exerts a force on body B, then body B exerts a force (equal in magnitude but opposite direction) on body A.
So: force on B from A = - force on A from B.
We have two forces on two different bodies.
So your conclusion may be: If the earth pulls on us, we pull on the earth (via gravity! Not via "pushing with out feets". Like the sun on the earth and vice versa). No sand at all so far. Well, we don't fall but the sand $brakes^1$ our motion, so it certainly exerts a force on us but less than the earth does on us! No one ever said, that the force of the earth on us is the same as the force of the sand on us (remember: the law speaks of two forces on two different bodies). On the other hand, it is of course true that we exert the same force as the sand exerts on us on the sand. That's why the sand moves away.
1: This is not the correct physical formulation; constant speed, no force, blabla. Of course, but the point of this answer is to make the distinction between on whom the forces act. And not on an exact description of a non-linear motion.
Two things.
A uniform motion (like sinking into quicksand with constant speed) does not require a net force; the force onto our foot soles touching the sand exactly counteracts our weight. If that was not so, we would accelerate. In fact, Newtons's First Law forbids a force difference. We accelerated when we started to sink in, and then there indeed was a net force (because the soft sand or fluid was giving and could not carry us) as is required for acceleration.
But what happens once we have started sinking in? In fluids and probably in quicksand (which I assume behaves similar to a fluid) the friction increases with velocity; at some point a body moving through it will encounter an amount of friction which cancels out its weight. From then on it will move with constant velocity because no net force is exerted on it any longer. For a person falling through air that's the terminal velocity, about 250 km/h. In water the friction is much stronger and the terminal velocity will be much smaller.
Interestingly there is an equilibrium of forces in the complete system even when we accelerate; the earth is drawn towards us by the same force by which we are drawn to the earth, and it accelerates (ever so minutely) towards us, preserving the earth+man system's momentum.
