All the following words are synonyms of "for example": For instance, e.g., Including, Such as, Like. Other creative ways to say "for example" are: To illustrate, Case in point, To show what I mean, To give you an idea.
In either case means in one case or the other. You can use it when you have only two options. In any case means whatever the situation is or will be. You can use it when you have three or more options.