should I implement the Monad, Applicative or Functor type class?
You rarely have to implement these type classes. But you need to understand how they work since many libraries use them. If you do IO, error handling, concurrency, use containers, option parsing and so on, you'll have to use these type classes.
For your own types, nobody forces you to implement them. If it turns you can make your type an instance of some type class, you may be able to reuse existing code rather than reimplementing it. And it will make the program more readable too.
Comments
You rarely have to implement these type classes. But you need to understand how they work since many libraries use them. If you do IO, error handling, concurrency, use containers, option parsing and so on, you'll have to use these type classes.
For your own types, nobody forces you to implement them. If it turns you can make your type an instance of some type class, you may be able to reuse existing code rather than reimplementing it. And it will make the program more readable too.