[RPG] the Fastest a Character Can Move in One Turn

I can improve on markovchain's answer by swapping a phantom steed for the wild-shaped druid. The phantom steed has a speed of 100 feet; we can improve this using the usual tricks (longstrider, transmuter stone, horseshoes of speed) to 150 feet, which means 15 miles per hour.

Like all creatures, the phantom steed can travel at +33% speed for a "fast pace" (this is what the line "or 13 miles at a fast pace" in the spell text means). This gives 20mph. markovchain offers two innovations: (1) an Elk Totem Barbarian can double this, and (2) the creature can "gallop" for one hour, which doubles this again. This gives 80mph.

Our phantom steed can gallop for only an hour, so we need eight phantom steeds to do this properly. If the DM rules that a horseshoes of speed can be reapplied to a new phantom steed very quickly, then our Elk Barbarian 6 can multiclass and take five levels of Transmuter Wizard, which will let her read the phantom steed spells from scrolls. She would need one scroll of longstrider and six scrolls of phantom steed in addition to her spell slots.

It's likely that the DM will rule that horseshoes of speed take significant time to put on a horse. (See: How long does your farrier take to shoe one horse?) If that's the case, then we need to assume the existence of courier stops, spaced every 80 miles along the route. Each courier stop would have a fresh phantom steed mount with the horseshoes already applied. Presumably a fifth-level wizard would supply the magic for each one.

80mph times eight hours gives 640 miles.

The above is a good practical solution. For completeness, I should note there are less practical solutions which are faster -- for example, with a line of friendly wizards, spaced 500 feet apart, with readied actions to cast dimension door on you, you can move arbitrarily far in one turn. There's also cheese involving fast mounting and dismounting, or even just a line of Giants with readied actions to pass you from one to the next.

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.