From D&D 3.5 to 5.0(5E) - 22 - Where Did My 5' Step Go? Disengage

It is gone.  D&D 5.0 doesn't have the idea of a 5' step.  But you don't need it for movement since you can always move 5', then do something like take an attack, then move some more.  Since 'attacks of opportunity', now known as 'opportunity attacks' still exist, you might want to consider finishing the first monster you attack before moving toward another foe.

You also can't use a 5' step to move out of an attacker's range for free.  If you want to stop fighting the enemy you are standing next to you need to use the "Disengage" action to prevent your enemy from making an opportunity attack.  In fact, while you are "Disengage" you can move past several enemies avoiding opportunity attacks from each of them.  The distance you can move on your turn during "Disengage" is the same as your movement rate.  So if you can move 30' on your turn, you can move a total of 30' on your turn while disengaging.  You can't disengage for 30' then move an additional 30'.