"diario" is also the one used at school by students to write down homework (eg. "studiare da pag. 11 a pag. 14"), communication from the school to parents and dates of the tests. The complete name, in this case, is "diario scolastico" but it is normally referred to using the word "diario" (teachers often say "prendete il diario e scrivete i compiti or "scrivete sul diario..." and students often ask "lo scriviamo sul diario?"). This kind of notebook starts with September and ends with the month of June, like school.
Then, "diario" also refers to a "diary" where you note down your feelings and thoughts (especially used by teenage girls or women). It may have different forms, even a notebook can be used as a diary.
"Agenda" can refer to a sort of daily planner, usually from Jan 1st to Dec 31st, to write down appointments and things to do.