Like any question, this answer will depend on the specific time that you are referring too. I'll do my best to answer what I can in a broad overview though.

The Rashidun Caliphs

The rise of Abu Bakr to Caliph following Muhammed's death in 632AD led almost immediately to religious based war -- The Ridda Wars -- which lasted until 633. Ridda (ردة) means apostasy in Arabic, and accordingly these campaigns were launched to try and bring the tribes surrounding the new Islamic Caliphate fully under Muslim dominion. Various tribes had, after the death of Muhammed, refused to pay the zakat, or 1/40th tax that Muslims are required to pay under the 5 pillars of Islam. The reasons for this varied. Some tribes saw the tribute as specific to Muhammed himself. Others didn't renounce Islam, but refused to pay tribute or taxes. Others too announced that they had their own prophets and religions. Abu Bakr saw all of these as a direct repudiation of Islam, and took action against those tribes. This was reflected in the orders that Abu Bakr gave to his commanders throughout the wars. They were to call the Adhan, or call to prayer, and see if the tribes they approached responded properly back with the adhan. The commanders were then ordered to ensure the tribes submitted to Abu Bakr, including the payment of the zakat. If the tribes did not submit, or even merely refused to pay the zakat, they were to be considered apostates.

This is slightly odd to some modern historians, such as the Sunni Historian Dr. Muhammad Hamidullah, who noted in his book Introduction to Islam that Muhammed himself had granted some tribes, such as the Ta'if, permission to not pay the zakat or take part in military service. This leads some to view these wars not as a religious war per se, but rather as the first Civil War that the Caliphate managed. In all truth, it could have been some of both. Many of the tribes that Abu Bakr dealt with, especially in the south near modern day Yemen and Oman, were openly hostile to Islam and claimed to follow their own prophets or leaders.

Anyways, that's a brief bit about the first action of the Rashidun Caliphs. Abu Bakr was waging war under the guise of Apostasy to make sure that the tribes stayed loyal to the Caliphate and paid the alms and tributes that they were meant to.

Treatment of other Religions in the Expansion of the Empire

One key aspect about the early expansion of Islam is that it did not attempt to force its way of life upon other people. Under the Caliph 'Umar, it was envisioned that Islam would be for the Arabs, who would exist as a sort of military caste above the conquered people. This meant that the conquered populations were disturbed as little as possible, and conversions were only on a voluntary basis. This precedent was set with Muhammed allowing the Jews and Christians of Arabia to keep their religion if they payed the Jizya (poll tax). This payment would make a member of the religions of the book - Jews, Christians, and later Zoroastrians - members of the protected class, or dhimmi. These people would be reserved from having to pay the aforementioned Zakat tax and from military service within the empire. This isn't to say that taxes wouldn't get absurdly high though, as Ira Lapidus notes that taxes on peasants could reach upwards of 50% of the value of thhe goods. It does however mean that the Muslim invaders were not hated. In fact, some Christian tribes had allied themselves with the Muslims against the Sassanians or Byzantines.

Oddly enough, the Religions of the Book flourished in the early days of conquest with Islams coming. Many previously repressed sects were able to abound, and apocalyptic messages were numerous. This changed though as time went on. 'Umar II (r.717-720) forbade Christians to hold powerful government positions. Other successive rulers forbade minority religions to ride horses, bear weapons, or show outward religious symbols in public. Repression came in waves, as one Caliph would crack down (Such as Yazis II destroying churches and animal sculptures) and then others relent. Keep in mind through all of this that from the 7th to the 11th century, upwards of half the worlds Christians lived under Islamic rule, yet they were not forced to exile or forced to convert by force (at least in large-scale).

Introduction of Arabic

For the first few decades of empire, records continued to be kept in Greek or Persian, as the Caliphs followed the policy of retaining the existing structures of the conquered, including their accountants. However, this started to change under the Umayyad Caliph 'Abd al-Malik and his son al-Walid. They started to translate documents into Arabic (a process that was vital to Islam throughout the empire, especially in the House of Wisdom which I wrote about here, along with contributions of Islam to science) and keep records in Arabic. This was also the regime that introduced the first minted coins of the empire, inscribed with the shahada, or Islamic profession of faith. This is when Arabic started to be spread throughout the empire, and when people would have started to see Arabic in more and more facets of life.

I really wish I could write more, but I have to study now! Feel free to ask any questions, and I'll do my best to answer them. I limited myself to the Umayyad's like you mentioned, but there was a constant state of flux throughout the different empires and different Caliphs.

Sources

A History of Islamic Societies by Ira Lapidus is a good book that I myself own. I would definitely recommend it to anyone looking for a good overarching view of Islamic progression. This was my primary reference for this post.

This talks some more about Abu Bakr and his campaigns if you are interested.