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 (Basic Rules, p. 73):
- 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.
Dropping to prone is found inside the movement phase rules, which means that it is used in your turn's movement:
You can drop prone without using any of your speed.
So consider it an action that is in your movement action, but uses 0 ft. So even if you use your full 30 ft (or however much movement speed your character has), you can still drop to prone at the end, because it costs 0 ft of movement.
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.
This means that dropping to prone can either be done on your turn's move action, or during the move action of the reaction from readying an action outside your turn. To do this, you'll give your DM a trigger for your reaction to occur, then go prone. You can even use your movement speed first and then go prone, or go prone and crawl up to half your speed, because dropping to prone uses 0 ft of movement.
And yes, you can use your turn's move, and then move again on your reaction.
