According to the PHB, page 189:
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
On 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 speed. You can always move, and taking the Dash action only helps you move faster.