I suggest you might like to think about a simple example, and really think it through. The idea is that you use what you have been told about Newton's third law, but don't regard it as a sort of mysterious utterance which you have to accept. Rather, regard it in the first instance as a hint or pointer; it is saying "look, notice this about the forces, this is a valuable thing to notice ..." and then by applying it to some simple examples you eventually come to understand it, and that is the moment when you can also accept it.
The simplest example is perhaps a collision when two objects A and B first approach one another, then interact, and then move away from one another. During the interaction, each is exerting a force on the other. Let $\bf F$ be the total force on A. Never mind for a moment what caused that force. Let $\bf G$ be the total force on B. Again, never mind for a moment what caused that force (I use the letter $\bf G$ just to have a different letter from $\bf F$). The equation of motion of each body is (Newton's 2nd law)
$$
{\bf F} = m_A {\bf a}_A
$$
and
$$
{\bf G} = m_B {\bf a}_B
$$
where I am assuming the masses are constant, and the accelerations are ${\bf a}_A$,
${\bf a}_B$ respectively. (I am not sure if you are familiar yet with writing Newton's second law as rate of change of momentum, which is in fact a better way to write it, so that is why I chose to talk about acceleration).
Now let's think about how those forces come about. The idea is that a force always comes about as a result of an interaction between two or more things. In the present example, if there is a third thing nearby, say a charged metal plate or a planet or something like that, then it can contribute to both the forces $\bf F$ and $\bf G$. But if there is nothing nearby, then the forces $\bf F$ and $\bf G$ are each caused only by the single interaction between A and B. And in this case the third law predicts
$$
{\bf G} = - {\bf F}.
$$
If we now put this fact into the two equations above, then we see that
$$
{\bf a}_A = \frac{1}{m_A} {\bf F}_A
$$
$$
{\bf a}_B = -\frac{1}{m_B} {\bf F}_A
$$
so we find that the two bodies must be accelerating in opposite directions: either away from each other (when they repel) or towards one another (when they attract). This is of course exactly what is observed, because Newton's third law is in fact a correct observation about the nature of the physical world.
In this example there is something which is not accelerating: the location of the centre of mass of the system. So your intuition that "it all balances out and nothing accelerates" is not completely untrue, it is simply that it applies to the overall centre of mass, not the individual parts A and B.
Finally, let's briefly look at a case where nothing is accelerating, such as a brick sitting on a table on planet Earth. In this case there is a gravitational force downwards on the brick and upwards on the Earth, pulling the two together. So that is one pair of forces:
$$
\begin{array}{rcl}
\mbox{gravitational force on brick } &=& -m g \\
\mbox{gravitational force on Earth } &=& m g
\end{array}
$$
where to write these equations in 1 dimension, I am taking the upwards direction as positive. If these were the only forces acting, then the brick and the Earth would accelerate towards one another. But in the scenario under discussion, we we also have an electromagnetic force where the brick meets the table, pushing them away from each other:
$$
\begin{array}{rcl}
\mbox{normal reaction force on brick } &=& N \\
\mbox{normal reaction force on table } &=& -N.
\end{array}
$$
These are again equal and opposite, as Newton's third law says.
When the brick first lands on the table, it may wobble or bounce, because $N$ might not be equal to $mg$. But when it settles down and comes to rest, then $N$ is equal to $mg$ so the total force on the brick is then $N - mg = 0$. Notice that in this final result the balance is between two different types of force both acting on the same object. This balance is not an example of Newton's third law! But the physics behind it does involve two applications of Newton's third law, as shown above.
