Your intuition is correct. The Dash action simply increases your movement, but doesn't actually increase your Speed.
If your speed were 30 ft. and you decided to Dash, you'd still only spend 15 ft. (of the gross 60 ft. of movement) to stand up from Prone.
If you had an item, effect, or class feature that increased your Speed (e.g. the Boots of Speed, which double your walking speed while active, or the Mobile feat), then standing from Prone would cost more, according to the rules as written.
You can't take both Dash and Ready actions
Normally you don't "prepare a reaction". You take the Ready action instead. See Player's Handbook, page 193, "Actions in Combat"
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 so that you can
act later in the round using your reaction.
Then you use you reaction (if still available) to actually perform the readied action when trigger occurs. You spend both your Action and your Reaction in this case. Therefore, you can't use your action to Dash after you declared a readied action, because it is already spent.
Unless you have an extra Action, or you're a Rogue
As notices by @NeilSlater, a Rogue has his Cunning Action feature, which allows him to use a bonus action for Dash. You can take the Ready action afterwards. However, this doesn't give you more movement than usual, since you just can Dash twice.
Also, you can't use "my turn ends" trigger
PH requires "perceivable circumstance" for a trigger:
First, you decide what perceivable circumstance will trigger your reaction.
Turns are game mechanics, they don't exist in the game world, hence, are not perceivable by the character, see Can you ready an action for "immediately before my next turn"?
You can move up to your speed using the Ready action
However, the question remains — what if you spent all your movement, and then take the Ready action to move 30ft more? Yes you can do this, providing your speed is 30 (or more) feet:
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
Ready action works like a "delayed Dash" in this case.
Best Answer
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.
Dropping to prone is found inside the movement phase rules, which means that it is used in your turn's movement:
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.
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.