A metal wire has free electrons which moves in a random manner in the absence of any cell connected across it.
When the ends of the wire are connected to a cell, the electrons begin to move from the negative to the positive terminal of the cell through the wire, during which they collide with the fixed positive ions and other free electrons of the wire due to which the speed decreases and direction of motion changes.
They again accelerate after each collision towards the positive terminal and suffer collisions with other positive ions and free electrons again and the process continues.
Hence, the electrons do not move in bulk with increasing speeds from one end to the other but drift towards the positive terminal. This depicts how the wire offers resistance to current through it.