According to the PHB, page 189:
###Your Turn
Your Turn
On your turn, you can move a distance up to your speed and take one action.
This means moving is not the same as taking an action. You may always move a distance up to your speed on your turn without taking any actions (barring special conditions — but let's not get into that!).
That you can always move on your turn without taking actions means there is no “move action”. Because why would there be, if you can always do this anyway?
##The Dash Action##
The Dash Action
In the PHB, page 192, it says:
When you take the Dash action, you gain extra movement for the current turn. The increase equals your speed, after applying any modifiers. With a speed of 30 feet, for example, you can move up to 60 feet on your turn if you dash.
The Dash action is the closest thing to a “move action” in 5e, but it doesn't allow you to move. It allows you to increase your movement. You can always move, and taking the Dash action only helps you move faster.
##Readying Movement##
Readying Movement
We've been talking about moving on your turn always being possible. But outside your turn, it is not so unless you take a specific action.
From the PHB page 193:
###Ready
Ready
[...]
First, you decide what perceivable circumstance will trigger your reaction. Then, [...] you choose to move up to your speed in response to it. Examples include [...] "If the goblin steps next to me, I move away."
There is something called a reaction which is also available to you. For the purposes of this question, we don't need to dwell on what are the specifics of a reaction, though.
You may do the Ready action to do something on someone else's turn that is normally only possible on your turn, expending your reaction. This includes moving. (Note: this Ready action is not being used with a Dash action, it is being used to just move on another creature's turn.)