I'm guessing your FBD looked something like this, where $F_1$ is the external force you apply. I'm assuming here that the top and bottom block don't slide relative to each other, so the forces at their junction ($F_2$) are equal and opposite.
The net force on the top block is the force you apply, $F_1$, minus the frictional force the bottom block applies to the top block, $F_2$:
$$ F_{top} = F_1 - F_2 $$
Because $F_1 > F_2$ the net force $F_{top} > 0$ and the top block accelerates.
Response to comment:
If the two blocks don't slide relative to each other then their accelerations must be the same so:
$$ \frac{F_{top}}{m_{top}} = \frac{F_{bottom}}{m_{bottom}} $$
We know that $F_{top} = F_1 - F_2$ and $F_{bottom} = F_2$, so:
$$ \frac{F_1 - F_2}{m_{top}} = \frac{F_2}{m_{bottom}} $$
and a quick rearrangement gives:
$$ F_1 = F_2 \frac{m_{top} + m_{bottom}}{m_{bottom}} $$
and since $m_{top} + m_{bottom} > m_{bottom}$ this means $F_1 > F_2$.
Basically $F_2$ is only accelerating the bottom block while $F_1$ is accelerating both blocks, so $F_1$ has to be greater than $F_2$.
Friction doesn't oppose motion, exactly. Frictional forces oppose relative motion of two bodies that are in contact. If two bodies in contact are moving relative to each other, then each body will experience a force in the opposite direction of its relative velocity with respect to the other body.
Static friction works the same way. It will act in whatever direction is necessary to keep two bodies from moving relative to each other, so long as the magnitude of the force required to do so does not exceed a particular limit determined by the normal force and the coefficient of static friction. In your example, it is necessary for static friction to push the top block to the right in order to keep the surfaces from moving relative to each other, and so that's what the static friction does.
And yes, the fact that frictional forces are in opposite directions is just a consequence of Newton's Third Law.
Best Answer
It is correct that the friction force from the ground opposes the applied force on the lower block (block b), but there will be no friction forces between the two blocks unless the lower block is set into motion. If the ground friction force is static friction, it will match the applied force $F$ for a net horizontal force of zero and there will be no force causing friction to arise between the blocks. Only if the maximum possible static friction force between the lower block and the ground is exceeded setting the lower block into motion, will there be friction between the blocks to oppose such motion. It is important to understand that static friction only exists in opposition to a net force that would act on the object in the absence of any friction.
The upper block will experience a friction force in the forward direction only if the lower block experiences a friction force in the backwards direction by the upper block. But as indicated above, there will be no backwards direction friction force on the lower block unless the maximum possible static friction force on the ground is exceeded so that the lower block accelerates forward.
That is correct.
That is correct, but only in opposing motion of the lower block that is actually occurring (sliding), not in preventing motion of the lower block from occurring. When the lower block is set into motion, the ground friction becomes kinetic, which is generally less than the static friction that initiated the motion, and the applied force $F$ on the lower block will be greater than the kinetic friction force, call it $f_{kG}$, from the ground on the lower block. Now there will be a net force on the lower block for static friction imposed by the upper bloc to the lower block, call it $f_{s-ab}$, to oppose. Then the net force on the lower block becomes:
$$F-f_{kG}-f_{s-ab}$$
And from Newton's 2nd law its acceleration becomes:
$$a_{b}=\frac{(F-f_{kG}-f_{s-ab})}{m_a}$$
The misconception originates from the original statement of what you say you learned in school. There is no static friction force acting on the lower block by the upper block's surface unless the lower block is set into motion, as I discussed above.
In closing I suggest in the future your first step should be to draw free body diagrams for the individual blocks. In this example, one where the system is stationary (i.e., the maximum possible static friction force between the lower block and ground is not exceeded). The other for when the lower block is set into motion (i.e., the maximum ground static friction is reached and friction becomes kinetic) and you want to determine whether the two block move together with the same acceleration, or the top blocks slides on the lower when the applied force us great enough, as in the example of the "table cloth trick" shown in this video: https://www.google.com/search?q=table+cloth+trick&oq=table+cloth+trick&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQABiABDIJCAIQABgKGIAEMgkIAxAAGAoYgAQyCQgEEAAYChiABDIJCAUQABgKGIAEMgkIBhAAGAoYgAQyCQgHEAAYChiABDIJCAgQABgKGIAEMgoICRAAGAoYFhge0gENMjA2OTAwMjFqMGoxNagCALACAA&sourceid=chrome&ie=UTF-8
Hope this helps.