Class change basics: Class abilities, mastery abilities, and exam requirements
Classes determine Movement, weapon access, and growth modifiers. Class abilities require the current class; learned mastery abilities can be equipped after switching. Only one optional exam proficiency is required.
Original video: “Fire Emblem: Fortune's Weave class change guide”; Bilibili creator Deerran Kan Youxi; https://www.bilibili.com/video/BV1MwaP6gEiB/
Class stats and growth modifiers
Changing classes alters a unit's Movement, usable weapons, and growth direction. Movement mainly depends on class, while other stats increase at level-up according to current growth rates.
Characters have personal growth rates, with classes adding modifiers. Even in the same class, 
Mu and 
Zarcone develop differently over time. Consider both the character's natural growth strengths and what the class adds.
Check weapons and proficiency bonuses before reclassing
Highlighted proficiencies on the class page correspond to usable weapons and movement types. Do not focus only on added Strength or Speed and discover afterward that your old weapon is unusable.
Classes also provide different proficiency EXP bonuses, affecting later training efficiency. Separate class abilities from mastery abilities, too.
Class abilities versus mastery abilities
For 
Shido, class abilities include two extra equipped Combat Arts and +5 sword Critical. Its mastery ability is 
Ever Vigilant, restoring HP after defeating an enemy with a critical hit.
Class abilities work only in the current class. Mastery abilities such as Ever Vigilant require filling the class EXP bar. Once learned, they can be equipped in the ability list even after changing classes.
Having entered a class does not mean having mastered it. If mastery is your goal, fill its EXP bar and then check that the learned ability is equipped.
Required and optional proficiencies
Speak to the Certification Exam official and pass the exam to gain class eligibility. Different tiers require corresponding certificates, with level and Renown requirements as well.
Proficiency requirements have required and optional categories. Required ones are mandatory; when several optional ones are listed, meeting any one is enough. My Mu has only Gauntlet D, satisfying one optional condition for 
Gladiator, yet still reaches a 100% pass chance.
After qualifying, returning to the same class does not consume exam resources again.
Match class changes to personal strengths and needs
Gold indicates a proficiency strength and purple a weakness. 
Catania is strong in swords, lances, and flying, and her personal ability needs Speed to work well. At the Beginner tier, compare 
Ornius Rider's growth and proficiency benefits.
Instead of asking only which class is strongest, consider the stats this character needs, the proficiencies they train well, and the weapons available now. Those questions make the next step easier to choose.