From my understanding and experience, all 4 skills are connected. However, there are two divides:
1. reading & writing
2. speaking & listening
From reading/studying text books as well as other material, material could be structured into those two areas mentioned above. I like to focus on speaking & listening with my Korean students but from time to time students require text to understand the use of language.
In a normal lesson, I like to focus on pronunciation and communication skills and then gradually introduce the reading as well as some writing exercises that focuses on vocabulary.
Nonetheless, there are ways to focus on the skills of reading. You could provide some text for students to read but there is more you can do with text to let students learn how to and improve their reading skills.
For listening, I like to use some songs or pre-prepared
suplementary listening activities that I have bought from the book shop nearby. This can include telephone calls, listening for detail, listening for gist etc etc.
For writing, I like to use a format for students to learn and then try to write using that format, say like a CV or filling out an application form.
Speaking is fun. Just bring some prepared board games or some role cards and let the students enjoy their lesson.
So therefore, I agree that the four skills of learning a language are all connected. Sometimes, a professional teacher is able to focus on particular skills to help assist students improve particular skills. If you have any questions regarding teaching some area of language skill please ask.