I think it is clear that Euler's religious faith played a big part in both his professional life and in his family life. We can see this by the way he interacted with others and by the way he dealt with adversity. Our knowledge of his religious beliefs comes from two of his publications: Letters of Euler to a German Princess on Different Subjects in Natural Philosophy and Defense of the Divine Revelation against the Objections of the Freethinkers. We will refer to these as [Letters] and [Defense].
Concerning the scriptures, he wrote
The Holy Scripture not only provides those who are seriously concerned with the improvement of their hearts with the most powerful means to that end, but that it also leads them in time to a greater knowledge of God. [Defense, XXIII]
Concerning God's sovereignty and man's free will, he wrote
We must acknowledge the government and providence of God, who having from all eternity (foreseen) all the counsels, the projects, and the voluntary actions of men, arranged the corporeal world in such a manner, that it brings about, at all times, circumstances which cause these enterprises to fail, or to succeed, according as His infinite wisdom judges to be most fit. God thus remains absolute sovereign of all events, notwithstanding the liberty of men, all whose actions, though free, are, from the beginning, part of the plan which God intended to execute, when He created this universe. [Letters, p. 383]
Concerning divine providence he wrote
To [Scripture] belongs in particular the doctrine of both general as well as particular divine providence, through which we can recognize that we can never get into any circumstance where God did not expressly place us according to his infinite wisdom and goodness, and can come to the firm assurance that not a single hair may fall from our head without the will of our heavenly Father. Now if only we were to ponder this doctrine with the appropriate attention and apply it to ourselves, we would submit to the will of God under all circumstances without difficulty and even with pleasure, and in this way attain true happiness. [Defense, XXVIII]
Concerning Christ, he wrote
It is therefore a settled truth that Christ is risen from the dead: since this is such a marvel, which could only be performed by God alone, it makes it impossible to cast any doubt on the divine sending of Christ into this world. Consequently, the doctrine of Christ and of his apostles is divine and since it is directed toward our true happiness, we can therefore believe with the strongest confidence all the promises which have been made in the gospel regarding this life as well as the one to come, and view the Christian religion as a divine work aiming at our spirituality. But it is not necessary to elaborate further on all this, since each one who is convinced only once of the resurrection of Christ cannot doubt any further the divinity of Holy Scripture. [Defense, XXXVI]
For further information on Euler's theology read The God Fearing Life of Leonhard Euler by Dale L. McIntyre .