Anna, Unfortunately, there are as many approaches to reading as there are students, e.g., Orton-Gillingham, Reading Recovery, Wilson, American Reading, IRLA, etc., to teach reading as there are students, and EVERY school district has a different approach. The school district usually dictates which approach you will use with your students, and programs change depending on the flavor of the day, i.e, which program is currently in vogue and being pushed by publishers. Being trained in a specific program is not a guarantee of being best able to help your students. Also, where you get your graduate degree will also determine how you are trained--my program at Providence College used the Reading Recovery model.
As a literacy/reading specialist you need to be knowledgeable about how to determine the students' reading strengths and weaknesses regardless of the program the school uses. You need know how to build upon the strengths and work to improve the weaknesses. Learning and using various reading/writing strategies (reading and writing are reciprocal processes which mean they work together) to help your students improve their skills. This is training that you will receive during your graduate degree fieldwork. Good luck!