Eu anoto todas as tarefas no caderno.

Anoto comes from the verb anotar, which means “to jot down” or “to note.” It implies briefly recording information.

• Escrevo (“I write”) is more general and can mean writing at length.
• Registo (from registar) often suggests making an official or formal record.

Here, anotar is the natural choice when you casually note tasks on paper.

Portuguese is a pro-drop language: the verb ending already indicates the subject.

• Anoto todas as tarefas no caderno. is perfectly grammatical without Eu.
• Including Eu adds emphasis or clarity (“I do it” as opposed to someone else).

In Portuguese, plural nouns generally require the definite article:

• todas is a pronoun meaning “all,” and it pairs with the article as to agree in gender (feminine) and number (plural) with tarefas.
• Dropping as (todas tarefas) sounds ungrammatical.

No is the contraction of em + o, meaning “in the” or “on the” when referring to a masculine singular noun.

• em o caderno → no caderno
  If the notebook were feminine (a agenda), it would be na agenda (em + a).

Not really.

• Anotar algo sobre o caderno could imply writing on top of its cover or surface, not inside.
• Em (in/on) is used because you’re recording the tasks inside the pages.

You can invert the verb and subject or add a question particle:

• Anotas todas as tarefas no caderno? (informal, using tu)
• Anota todas as tarefas no caderno? (informal, você form)
• Você anota todas as tarefas no caderno? (full form, more neutral)
  You could also add Não é? or Certo? at the end for confirmation:
• Você anota todas as tarefas no caderno, certo?