I am wondering how this informal grammar works.
For example, 頑張る、頑張れ、頑張ろう - they all mean try hard, but how are they different in the tone of voice? In what situation should I use れ and ろう?
Thank you very much~
頑張る: To try your hardest.
頑張ろう： Lets try our best
頑張れ: (Like an order or a command almost) Try your best
They can be used in any scenario if you use them properly. But in the simplest use cases:
私は頑張る: I will try my best (this is plain form, so its informal. Changing it to 「頑張ります」makes it polite.
お互いに頑張ろう： Lets both try our best (quite informal).
Malina、頑張れ！！: Try your best, Malina! (very informal, but its the kind of thing you shout at your favourite sports players or close friends for example).
Of course there are many many complexities to this which would require another question as it's not necessarily specific to「頑張る」.
頑張る - this is known as the Dictionary form (lemma) and is the 'base' form of the verb. This form is used to express present and future actions.
頑張れ - this is the imperative form and is used to give commands or exhortations.
頑張ろう - this is the Volitional form and is used in several different constructions which contain a sense of intention. One common example is Volitional Form + とする to indicate 'try to'.
These forms are a little different depending on whether the verb is a godan verb or an ichidan verb. I recommend you review some basic textbook on verb grammar. It will explain these and other basic forms which comprise the foundation of Japanese verbs.