How can a massless string move if the net force on it is always zero? Lets say we have a frictionless and massless pulley with a massless string over it. There are two masses attached to it $m_1$ and $m_2$ where $m_2>m_1$.
So $m_2$ would an experience an acceleration down while $m_1$ experiences an acceleration up and massless string would also move accordingly.
However, if the net force on a massless string is always zero, it shouldn't experience any acceleration and stay at rest. But the string does move. How does it move?
 A: Newton's second law is
$$ F=ma $$
For a massless object with no force on it, this equation becomes
$$ 0=0*a $$
which is clearly true for any value of $a$, and so any value of acceleration is consistent with Newton's second law. Indeed, if the force was non-zero the acceleration is undefined. A massless object can have any acceleration, but it can never have a net force applied to it.
A: Your problem arises from the negligence of the fact that when we say massless, frictionless, etc. then we are only talking about the limiting case of these quantities tending to zero and not the absolute zero.
This in the given scenario means that you can neglect the mass of the string. But that doesn't imply that the string doesn't have a mass.
You might ask why mass cannot be zero (absolute one) here? Because force is the cause of acceleration and without any net force no body can accelerate, whether it be massless or not. As soon as the net force stops acting  on a body the acceleration of the body ceases.
A: The fact that the question states an assumption of massless pulley and string is to “idealise the situation”. Lots of physics problems ask you to ignore factors that are not particularly  relevant to the point of the question. Here, you are asked to consider the forces and motion of the masses only. From there you are to determine the motion of the masses. If the question stated otherwise then this would create another level of complexity which is not consistent with level of physics you are studying and the physics being currently taught to you.
But to answer your question, the string is attached to a mass and the mass experiences a force (and therefore acceleration) meaning the string will accelerate as well.
