Which leg should my first learn to walk with when learning to walk. Should be perfect using the right leg before trying to use the left? I want to learn to eat with chopsticks. Should I try to learn to eat with one stick first, and perfect that, as two sticks are more complicated?
Clearly, neither of the above situations is logical, and it is the same with your question.
You cannot learn one part without learning the other parts in conjunction. Most books and language courses for beginners try to teach reading, writing, speaking and listening, together. Courses typically contain one or two exercises from each skill per unit. When you miss out one segment, you create a gap in your skill set that will significantly limit your language potential later.
I see this happening commonly in China, where due to exams concentrating on reading and listening, the students study predominately reading and listening. These students study only a little writing and often completely ignore speaking.
Later in life, they encounter a situation where they have to use English, such as going abroad to study at university, in their employment, maybe they have a foreign visitor or a foreign business trip. At this point, the language gap becomes very noticeable.
Correcting the missing skill, however, becomes problematic. Late in life, it is difficult to study. They are missing the learning environment and teachers. They have little time to study with other commitments such as work and family. Going back to studying also means that they have to re-learn lots of basic sectors of the language, going over quite dull and boring learning tasks.
It is better to learn in a comprehensive way from the beginning.