There are never neutral atoms in pure $\ce{NaCl}$ or in its solutions. The pure $\ce{NaCl}$ crystal is made of piles of ions $\ce{Na^+}$ and $\ce{Cl-}$. When $\ce{NaCl}$ is dissolved into water, the ions $\ce{Na^+}$ and $\ce{Cl-}$ are separated. They become independent and can move in water. If you dip in this solution two conducting pieces bound to the poles of a battery, a positive pole (anode) and a negative pole (cathode) are created in the solution. The positive pole (anode) attracts the negative ions (anions) like $\ce{Cl^-}$ : they slowly migrate in the solution up to the anode, and when they touch the anode, they lose their charge and create a free electron which enters the anode and the external circuit. The anions become neutral atoms $\ce{Cl}$. So the anions do not take up electrons as you say. They already have electrons in excess, before the beginning of the electrolysis.
In the circuit "Negative pole of the battery - Electrode - Solution - Electrode - positive pole of the battery", the electric current is made of charge carriers which are electrons between battery and electrodes, and ions in the solution between the electrodes. The nature of the electric currant changes along the circuit.
To be correct I should also mention that positive ions like $\ce{Na^+}$ are also migrating in the solution, as they are attracted by the negative pole (cathode). So in the solution, there is one currant of positive ions moving to the negative electrode. And there is also a currant of negative ions moving to the positive electrode. As a result the solution remains neutral, without any positive or negative electric charges in excess.