I want to know what is the methodology of the Righteous Predecessors? What are the books that could help me understand them? Were the Salaf considered one of the movements that claim to adopt the true Islam? What are their schools of Fiqh?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.

The Salaf (righteous predecessors) refers to the Companions, the Taabi‘is (the succeeding generation; Followers) and those who followed them in the best way. The Quran and the Sunnah (Prophetic tradition) are their methodology and their followers are spread in every place and time. Their unconditional affiliation is to the Sunnah. It is important to mention that the methodology of the Salaf (righteous predecessors) is not exclusive to a particular movement that calls or attributes itself to the Salaf. Rather, any person who adopts their belief and acts according to their methodology belongs to them.

This is what is said about the methodology of Ahlus-Sunnah in issues related to creed.

As for Fiqh (Islamic jurisprudence), Ahlus-Sunnah follow the four Sunni schools of Fiqh, namely, the Hanafi, Maaliki, Shaafi‘i and Hanbali schools. The principles of these four schools were preserved and accurately reported from their respective Imaams. However, other prominent Sunni Imaams have their own schools of Fiqh, but were not completely recorded and compiled  like those of the Imaams of the four schools of Fiqh. Some examples are Al-Layth, Al-'Awzaa‘i, Sufyaan At-Thawri, Sufyaan ibn ‘Uyaynah and many others all. Nevertheless, the one who studies a particular school of Fiqh should avoid bigotry that is prohibited in Islam. Instead, he should adhere to the truth wherever it is found.

Allaah Knows best.