These words have similar meanings. In fact, in many non-English languages there are no separate equivalents for these three words.

Soon

Soon means ‘a short time after now’.

Get well soon. (NOT Get well early.)
Reply soon.

Soon can also mean ‘a short time after then’.

Driving in London was hard at the beginning, but soon she got used to it. (NOT Driving in London was hard at the beginning, but early…)

Early

The adverb early means ‘near the beginning of a period of time’. It can also mean ‘sooner than usual’ or ‘sooner than others’. Early does not usually mean ‘a short time after now’.

I usually get up early in the morning. (NOT I usually get up soon in the morning.)
We take our holidays early in the year.
The train arrived thirty minutes early. (= sooner than usual)
I had to wait because I arrived early. (= sooner than others)

Early can also be used as an adjective.

We expect an early reply.
I caught an early train.

A watch or clock can be fast or slow, not early or late.

My watch is ten minutes slow.

Quickly

Quickly refers to the speed with which something is done.

You are eating too quickly. (= You are eating at a high speed.)
She quickly got ready.
Come and see us soon. (= Come and see us before too long.)
Come and see us quickly. (Make the arrangements fast)