Short Answer: Just falling to prone counts as movement and still requires to be on your turn, but it's something that can easily be setup as a reaction.
There are another list of small, seemingly insignificant actions that can be rolled up into your actions. This list is located just below the Being Prone rules:
INTERACTING WITH OBJECTS AROUND YOU
Here are a few examples of the sorts of thing you can do in tandem with your movement and action:
draw or sheathe a sword
open or close a door
withdraw a potion from your backpack
pick up a dropped axe
take a bauble from a table
remove a ring from your finger
stuff some food into your mouth
plant a banner in the ground
fish a few coins from your belt pouch
drink all the ale in a flagon
throw a lever or a switch
pull a torch from a sconce
take a book from a shelf you can reach
extinguish a small flame
don a mask
pull the hood of your cloak up and over your head
put your ear to a door
kick a small stone
turn a key in a lock
tap the floor with a 10-foot pole
hand an item to another character
These are all small actions, but still require it to be your turn and using (even if it is in tandem with) your actions to perform.
So simply dropping to prone requires a small but at least some effort, which is used in your movement phase. Now, I'm glad you brought up the Ready action, because that is exactly what you can do with it.
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.
First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it. Examples include "If the cultist steps on the trapdoor, I'll pull the lever that opens it," and "If the goblin steps next to me, I move away."
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger. Remember that you can take only one reaction per round.
With a reaction, you can give the DM a trigger, and say that you will drop prone when this happens. This is helpful because since dropping to prone uses 0 movement, you can also use your reaction movement to jump back up from prone, or to even crawl away.
And yes, you can use your turn's move, and then move again on your reaction.