Revelations in Mecca (610-623). At this time Islam was a pursued religion what taught three main doctrines:

• Monotheism: returning back to the way of Abraham, the father of all Monotheistic faiths.
• Prophecy: Accepting every Scriptures, Prophets, Angels, Judgement day which are based on Abraham’s Monotheism. Accepting that Mohammed (PBUH) is the last Prophet in the sequence of the prophetic line and the Quran is the last revelation summing up and clarifying all the previous ones.
• Belief in the Hereafter and final reckoning.

All the three teachings are dogmas what nobody can dispute however they meet the basic doctrines of the previous monotheistic religions. In different ages and places the Islamic thought resulted different schools and movements.
In the Islamic understanding of Islam doctrines and sharia cannot be divided. Only both of them together can be considered Islam. For Muslims freedom of faith means freedom of practicing Islamic sharia as well.

The other period is related to Medina (623-633), where the first Islamic State was formed and a legal system was established in connection with it. This laid the foundation for the formation of schools of jurisprudence.
I will describe the schools of thought below.

Traditional schools referring to Wikipedia:

Kalam

Kalam is the Islamic philosophy of seeking theological principles through dialectic. In Arabic, the word literally means "speech/words". A scholar of kalam is referred to as a mutakallim (Muslim theologian; plural mutakallimun). There are many schools of Kalam, the main ones being the Ash'ari and Maturidi schools in Sunni Islam.

Ash’ari

Ash'arism is a school of theology founded in the 10th century by Abu al-Hasan al-Ash'ari. The Asharite view was that comprehension of the unique nature and characteristics of God were beyond human capability.

Maturidi

Maturidism is a school of theology founded by Abu Mansur Al Maturidi, which is a close variant of the Ash'ari school. Points which differ are the nature of belief and the place of human reason. The Maturidis state that belief (iman) does not increase nor decrease but remains static; it is piety (taqwa) which increases and decreases. The Ash'aris say that belief does in fact increase and decrease. The Maturidis say that the unaided human mind is able to find out that some of the more major sins such as alcohol or murder are evil without the help of revelation. The Ash'aris say that the unaided human mind is unable to know if something is good or evil, lawful or unlawful, without divine revelation.

Traditionalist theology (Hadith)

Traditionalist theology, sometimes referred to as the Athari school, derives its name from the word "tradition" as a translation of the Arabic word hadith or from the Arabic word athar, meaning "narrations". The traditionalist creed is to avoid delving into extensive theological speculation. They rely on the Qur'an, the Sunnah, and sayings of the Sahaba, seeing this as the middle path where the attributes of Allah are accepted without questioning their nature (bi la kayf). Ahmad bin Hanbal is regarded as the leader of the traditionalist school of creed. The term athari has been historically synonymous with Salafi. The central aspect of traditionalist theology is its definition of Tawhid, meaning literally unification or asserting the oneness of Allah.

Murji’ah

Murji'ah was a name for an early politico-religious movement which came to refer to all those who identified faith (iman) with belief to the exclusion of acts.

Qadiriyyah

Qadiriyyah Qadariyyah is an originally derogatory term designating early Islamic theologians who asserted that humans possess free will, whose exercise makes them responsible for their actions, justifying divine punishment and absolving God of responsibility for evil in the world. Some of their doctrines were later adopted by the Mu'tazilis and rejected by the Ash'aris.

Mu’tazili

Mu'tazili theology originated in the 8th century in al-Basrah when Wasil ibn Ata left the teaching lessons of Hasan al-Basri after a theological dispute. He and his followers expanded on the logic and rationalism of Greek philosophy, seeking to combine them with Islamic doctrines and show that the two were inherently compatible. The Mu'tazili debated philosophical questions such as whether the Qur'an was created or eternal, whether evil was created by God, the issue of predestination versus free will, whether God's attributes in the Qur'an were to be interpreted allegorically or literally, and whether sinning believers would have eternal punishment in hell.

Jahmiyyah

Jahmis were the alleged followers of the early Islamic theologian Jahm bin Safwan who associate himself with Al-Harith ibn Surayj. He was an exponent of extreme determinism according to which a man acts only metaphorically in the same way in which the sun acts or does something when it sets.

Bateniyyah

Bateniyyah The Batiniyyah is a name given to an allegoristic type of scriptural interpretation developed among some Shia groups, stressing the baṭin (inward, esoteric) meaning of texts. It has been retained by all branches of Isma'ilism and its Druze offshoot. The Alawites practice a similar system of interpretation.