How exactly do accelerated charges produce electromagnetic waves So I know that accelerated charges produce changing magnetic fields which in turn produce an electric field, according to Faraday's law.
(Of course keeping in mind that the electric and magnetic fields are perpendicular to each other)
What after that? How is the electromagnetic wave created?
 A: Charges affect each other at a distance after a time delay, whether they are moving or accelerating or not.
When both are stationary we consider the force an electric field.
When the target is stationary and the source is moving directly toward or away from the target at constant velocity, it's still an electric field but the intensity of the force is different.
If the target is stationary and the source is moving sideways, it's still an electric field and the intensity is something else.
If the source and the target are both moving, and the source has some motion sideways, while the target has some motion in the same plane as the direction between them and the source velocity, then there is also a magnetic force. In this particular case, the equations for the electric force do not add up in different frames. In a frame where one charge is stationary magnetic force on the target is zero. In a frame where they are both moving, the equations give a different wrong result and magnetic force is needed to make up the error.
You get radiation only when the source charge is accelerating sideways. 
The radiation will be a wave pattern when the motion of the source charge fits a wave. 
A: 
So I know that accelerated charges produce changing magnetic fields which in turn produce an electric field, according to Faraday's law. 

It is the "in turn" that is not correct, the law cannot apply to light as light  has no charges. Look at this animation according to the  mathematics of Maxwell equations:


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. The electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together

The solutions of Maxwell's equations describe the behavior of classical electromagnetic waves. Charges have electric fields, moving charges create magnetic fields , and these   observations become  the laws,  entering Maxwell's theory axiomatically . It is a full semester course to understand classical electrodynamics in all its complications. Yes, accelerated charges produce electromagnetic waves, as the one shown in the annimation.
To understand how the classical wave is created from the underlying quantum mechanical framework of nature requires studying both quantum mechanics and quantum electrodynamics, QED. More than semester courses.
It can be shown that the classical electromagnetic  wave emerges from a confluence of photons, the photon is  quantum of light, not in an additive way, but in a superposition of the quantum mechanical wavefunctions of the photons. 
