First of all, Snoke and the Knights of Ren are NOT Sith. This will be important later in the answer.
When one is said to be "ready" to "complete his training", it usually means the moment when one is ready to complete his/her final stage of apprenticeship. In this case it refers to the moment when a Jedi is ready to undergo the final trials to qualify for Knighthood, or when a Sith apprentice is ready to become a master. Generalising, any Force user's trials involves testing his/her ability to stay true to his alignment in the Force and readiness to advance the goals of his/her chosen order. The trials can thus be broken down into two key components:
- The ability to overcome and resist the temptation of the dark side, and master the emotions that lead to it (for light-siders) or to embrace said emotions and the power of the dark side while resisting the call of the light and, in effect, overcoming one's moral conscience and "humanity" (for dark-siders)
- The readiness to serve as a guardian of the galaxy (for light-siders) or to advance the cause to wipe out the Jedi Order and any other light-siders first and foremost, and conquer the galaxy second (for dark-siders)
The above is the general theme that can be inferred out of all Force users in canon so far who is deemed "ready".
The Jedi Order during the Galactic Republic era has a formalised system of trials to testing Padawans. The Five Trial system you quoted belongs to Legends, not canon. In Disney canon, the Jedi Trials consist of nine steps, each testing the Padawans ability to stay true to the Jedi ways in nine aspects:
teamwork, isolation, fear, anger, betrayal, focus, instinct, forgiveness, protection.
Since you've quoted it, I'll also elaborate on Legends' Five Trial system. The standard test during the Republic's Golden Age (the era when the Sith are thought to be no more) and acceptable alternatives to bypass the corresponding trial is given.
- Trial of Skill: Avoiding distraction through self-discipline while engaged in lightsaber combat with a designated opponent and battlefield, emerging victorious. Bypassed if the Padawan bested a battlemaster during a sparring session or achieved "blindfolded mastery of the saber training exercise Faalo's Cadences".
- Trial of Courage: A demonstration of courage in dangerous missions, either simulated or actual. Death is entirely possible. Bypassed in special circumstances on a case-by-case basis. Examples include overcoming a crisis that involves saving his/her Master's life, or surviving a Sith battle encounter.
- Trial of the Flesh: The overcoming of great pain or loss, physical or emotional. Said pain is frequently applied during the test itself - bloodshed is common. Bypassed in cases where Padawans overcame existing pain/loss of an acceptable level. Examples include Padawans who experienced (and remembered) loss of family prior to joining the Order and successfully overcame those emotions.
- Trial of the Spirit: Facing the Mirror - facing and overcoming the darkness within themselves through meditative self-discovery. Potentially extremely dangerous, the journey is beyond the control of any Jedi Master. It is possible for one to lose his/her sanity. Such trials are always supervised by a Master, whose role is to guide you to the darkest depths of your mind and back, but not defining its details. Bypassed if the Padawan demonstrates the same mastery of their own dark side by other means.
- Trial of Insight: The final test - The ability to seek, through the Force, the truth that is hidden from one's physical senses. Puzzles stored in the Jedi archives are randomly selected. Key elements involve the use of the Force to see past illusion, disguise, lies etc. Bypassed through the demonstration of wisdom beyond his/her years - especially in the divination of solutions to successfully avoid violence.
The Sith Order of the Galactic Republic and Empire era operates according to the Rule of Two:
Always two there are. One to embody power, the other to crave it.
Two Dark Lords of the Sith: One a master, the other an apprentice. When the apprentice has learnt all he could and is strong enough to kill the master in a duel to the death, he is deemed ready to take his master's place as supreme lord of the Sith. This philosophy is best explained by Palpatine himself, the final culmination of the Rule of Two at its pinnacle of success (cr: Lords of the Sith):
Soon after destroying the Jedi, the Emperor had told Vader that he would one day be tempted to kill him. He’d said that the relationship between Sith apprentice and Master was symbiotic but in a delicate balance. An apprentice owed his Master loyalty. A Master owed his apprentice knowledge and must show only strength. But the obligations were reciprocal and contingent. Should either fail in his obligation, it was the duty of the other to destroy him. The Force required it. Since before the Clone Wars, Vader’s Master had never shown anything but strength, and so Vader intended to show nothing but loyalty. In that way, their mutual rule was secure. Perhaps Vader would attempt to kill his Master one day. Sith apprentices ordinarily did. They must, if they were trained well. An apprentice was unquestioningly loyal until the moment he wasn’t. Both Master and apprentice knew this.
In other words, once the master has taught all he can, and the apprentice equals the master in mastery of the dark side - in effect, he has grown to be a master himself - then the two must battle to determine who is the greater master, the lesser to be slain, for there can only be one of each role at any time.
As for Kylo Ren, he is not a Jedi nor a Sith. So the trials listed above do not apply. There's no standardised test we know of, nor any master to kill. However, his readiness to "complete his training" still follow the same two principles listed above: Extinguishing the light within him and demonstrating the ability to exterminate the Jedi. As for what Snoke means by the final training...we'll know in Ep IIX, hopefully.