Germany was still a deeply religious country by the time the Nazis came to power, 95% of the population was some sort of christain denomination. The nazis were a diverse group when it came to religion. Himmler and Rosenberg subscribed to their neo-pagan Germanic beliefs, there were radical atheists, and there were some who still held onto some sembelence of their former religion. The Catholic Church and the nazis had signed a concordat in 1933 outlining their relationship and more importantly guaranteeing the position of the Catholic Church in Germany. The Catholic Church was heavily opposed to both the killing of the Jews and the euthanasia program run by the nazis. The deal the papacy made with the nazis was in tatters. The nazis closed down church organizations and charities, and arrested many clergy men. The Catholic Church under pius xi issued a number of documents heavily condemning the nazis. There was a propaganda war of sorts between Germany and the church. Church leaders, like a famous priest named Galen, would give sermons denouncing the euthanasia programs, the nazis, unable to kill Galen lest he become a martyr, released propaganda films showing euthanasia as just. Hitler eventually broke all relations with the papacy and began to seriously repress the church in 1942. He declared that Christianity will suffer a "slow death" as it gets worn away before the advances of science.

Now the relationship with Protestants was a bit more complicated as many Protestants were supportive of the nazis and hitler wasn't nearly as oppressive when it came to Protestant churches. A famous incident called the "Bethel hospital" incident soured relations between the nazis and Protestant churches. What happened was a Protestant bishop, famous worldwide for his charity work, refused to give up his handicapped patients, the nazis responded by bombing the church. The nazis replaced many of the priests and bishops with pro-nazis bishops, this coupled with internal squabbling led to the Protestant church largely falling under nazis domination. There were still plenty of outspoken critics that came from the Protestant denomination, but many supported the Nazis and agreed with the racial purity initiative that the nazis pushed. Some churches even issued decrees supporting the policies of the nazis.

Now as for Islam, hitler generally respected their warrior like origins, and he used Muslims as tool to help fight the British in Africa and he used Muslim as recruits to preform partisan operations in the Balkans. I think it goes without saying what he though about the Jewish faith.

Source:

The third reich series by Richard Evans

Mein Kampf by Adolf Hitler

First, though some posts have mentioned the Reichskonkordat, it may be helpful to have a translation of the document. There have also been statements regarding that “later” the relationship between the Vatican and Germany were regulated by the treaty (which is what Concordats are—bilateral treaties between the Vatican and another nation). As Hitler was appointed chancellor in 1933 (January, as I recall) and the Reichskonkordat was signed in July of 1933, there was little time for relations to change between Hitler coming into power and the Concordat’s application. Violations of the treaty started nearly immediately, and dozens of protests from the Vatican between 1933 and 1939 fell on deaf ears in Berlin. Pius XI was the pontiff during this period. His Cardinal Secretary of State was Eugenio Pacelli, who was the Vatican official who signed the treaty and would also be elected pope in 1939 (and would take the name Pius XII). Fritz von Papen was the German signatory.

The most direct opposition to Nazi ideology was provided by Pius XI in his encyclical Mit Brennender Sorge. The full text in English can be read here. This document, released in March of 1937, was a direct challenge to Nazi ideology. It had to be smuggled into Germany, and after it was read to Catholic congregations the Gestapo attempted to seize as many copies of it as possible. Subsequently, Vatican Radio and Catholic newspapers were banned.

The first part of the encyclical deals with condemning Nazi violations of the Reichskonkordat, signed in 1933. The remainder deals with outlining the problems the Church had with Nazi ideology. Perhaps the most striking excerpt comes from section 8:

Whoever exalts race, or the people, or the State, or a particular form of State, or the depositories of power, or any other fundamental value of the human community - however necessary and honorable be their function in worldly things - whoever raises these notions above their standard value and divinizes them to an idolatrous level, distorts and perverts an order of the world planned and created by God; he is far from the true faith in God and from the concept of life which that faith upholds.

It may be useful at this point to discuss some peculiarities in encyclicals. At their most basic, they are letters from a pontiff to a particular bishop or area of the world. However, they have been leveraged by the papacy since the 1800's as a teaching tool and a way to address issues in the world. While they do discuss events going on at the time of their publication, they are not focused only on the present. Encyclicals are used to address the present, the past, and the future--often in equal measures. As such, their language is inherently biased toward generalizations over specificities. In this case, rather than merely addressing Nazism in Germany, Pius XI chose to speak about any possible governments--past, present, and future--that exhibit similar tendencies. So you will not see the term 'fascism' in Mit Brennender Sorge. Context is critical to understanding encyclicals, and without proper context they can appear to be relatively bland statements. (Basically, if you're looking for the evocative and fiery prose of Churchill, you're going to be disappointed) The nature of encyclicals is actually addressed in section 41:

We have weighed every word of this letter in the balance of truth and love. We wished neither to be an accomplice to equivocation by an untimely silence, nor by excessive severity to harden the hearts of those who live under Our pastoral responsibility; for Our pastoral love pursues them none the less for all their infidelity

Returning to the text, sections 15 and 16 praise the Old Testament, an oblique statement of support for German jews. This becomes more explicit in section 18:

The Church founded by the Redeemer is one, the same for all races and all nations. Beneath her dome, as beneath the vault of heaven, there is but one country for all nations and tongues; there is room for the development of every quality, advantage, task and vocation which God the Creator and Savior has allotted to individuals as well as to ethnical communities

Another explicit refutation of the ideology of race is present in section 11:

None but superficial minds could stumble into concepts of a national God, of a national religion; or attempt to lock within the frontiers of a single people, within the narrow limits of a single race, God, the Creator of the universe, King and Legislator of all nations before whose immensity they are "as a drop of a bucket" (Isaiah xI, 15).

The call to action by individual Catholics is in section 19, "at no moment of history, no individual, in no organization can dispense himself from the duty of loyally examining his conscience, of mercilessly purifying himself, and energetically renewing himself in spirit and in action." It is clear, sadly, that many Catholics had divided loyalty and did not heed the call to heroism in section 21. Far too many chose country over Church in this instance, and many Catholics participated in the atrocities of WWII.

This encyclical was one of three issued in March of 1937. Divini Redemptoris dealt with the Church's opposition to Communism, and Nos Es Muy Conocida addressed the situation in Mexico. Taken as a whole, these three encyclicals constitute Pius XI's attempt to hold back the brewing conflicts of the 20th century and the ongoing tragedy of the Spanish Civil War. In each case, the reaction among Catholics was mixed. The appeal of loyalty to your nation over your faith was a strong one, and resisting the power of a nation required exceptional heroism. There were Catholic heroes, but there were also Catholic villains. There was, however, official resistance to Nazi ideology from the papacies of Pius XI and Pius XII.

You may also be interested in my post regarding Pius XII and allegations of his silence during the Holocaust.

As always, followup questions from OP and others are highly encouraged.

The Nazi government had a mostly hands off approach to religion, with southern Germany & Austria mostly Catholic and northern Germany Protestant they thought an open opposition to religion would add further reason for resistance to form(Although during the early years they faced opposition by Catholics, although this was later resolved thanks to the Reichskonkordat). In Nazism the ultimate loyalty of every man, woman, and child in Germany should belong to the state and the national socialist ideology. Christianity and religion period made this difficult. Although they did and had plans to subvert Christianity by trying to fuse Christian views and practices with Nazi ideology. On the surface Nazism was pro-Christianity, but under the surface it was fundamentally opposed to it and most organized religions.

Hitler was manipulative, his religious views reflected this. He praised Christianity in public and used it when it suited his needs, but as far as we know he was personally an atheist. In private he was known to criticize Christianity as a whole and despised organized Christianity. He eventually wanted to get rid of Christianity in a subtle way as he and other Nazis saw it fundamentally against the beliefs of National socialism. But he also opposed the paganism promoted by Himmler. I believe he had favorable views of Islam however. So Hitler's personal religious beliefs lay strongly on the atheist side, with some anti-theist leanings.

Edit Sources:

• Goebbels Diaries by Joseph Goebbels
• Hitler a Study in Tyranny by Alan Bullock
• Hitler and Stalin: Parallel Lives by Alan Bullock
• Inside the Third Reich, the memoir of Albert Speer