You might want to look at the FAQ for Nazis and the Occult, which answers some of your questions, but there are a few points that can be addressed here:

So from what my dad has told me, the Nazi "elite" were originally a part of some cult that saw themselves as descendants of Aryan gods and tried to find some mystic artifacts throughout their reign of Germany.

The Occult Roots of Nazism : secret Aryan cults and their influence on Nazi ideology : the Ariosophists of Austria and Germany, 1890-1935 (1985) traces some of the origins of National Socialism to German esoteric movements, and suggests that some members of the Nazi regime were interested in the occult or esoteric theories, particular Heinrich Himmler, and others like Rudolf Hess were members of the Thule Society (Thule-Gesellschaft). It's important to remember that at this point (1920s-1930s) archaeology and anthropology were still fairly young sciences, and very much caught up in scientific racialism and euhemerism (i.e. trying to find the "truth" behind folklore and legend), and a number of occult, pseudoscience, and "fringe" movements were very prominent, alternately borrowing from the new publications on anthropology and positing their own theories - like the Theosophists, who took a mishmash of Indian legendry, Western occultist, and 19th century racial concepts and combined it into an extensive mythology. So the idea that certain people among the Nazis were interested in the occult, or subscribed to fringe beliefs, is no stranger than someone being a Scientologist today. But the idea that the "Nazi elite" as a group were interested in the occult, or part of a cult, is an idea that was really developed and pushed after the war. Case in point...

He said they sent out search parties to look for "The Holy Grail", "The Ark", and they apparently already found the "Spear of Destiny".

This is an idea that was popularized by Raiders of the Lost Ark and Indiana Jones and the Last Crusade, Constantine, Hellboy, and books like The Spear of Destiny: the occult power behind the spear which pierced the side of Christ. The truth is buried in two separate actions: first, the Nazis under Hitler did raid the museums and private collections of continental Europe for artwork and historical relics, including one artifact that is claimed to the "Holy Lance" (also known as "the Lance of St. Maurice") that was in Vienna, but taken to Nuremburg, and was returned to Austria after the war.

Second, the Nazis engaged in archaeological and anthropological efforts as an extension of state propaganda, to support German racial claims. The most well-known of these organizations was the Ahnenerbe, which was headed by Heinrich Himmler, and the operations they undertook were of questionable scientific purpose, and some of the theories they promoted were ones that met the party's ideology rather than the evidence. So for example, there was a 1938 Ahnenerbe expedition to Tibet, led by Ernst Schäfer. Expeditions to the Middle East to recover legendary Biblical relics are fictional, however.

These all seem to be artifacts from Abrahamic Religion, but not only is he insistent that this isn't about religion, but he mentioned that these Norse Aryan race that the Nazi elite thought they were a part of was traced back to the Babylonians. I'm terrible at history, but didn't Babylon come before Abrahamic religions and did they ever have anything to do with them?

Nazi ideology held that the German people were the descendants of the Aryan race, who spoke the Indo-European language which linguists believe is the precursor of most European and Indian languages. The idea of a conquering Aryan race of white people appealed to a lot of people at the time. Most pro-Aryan race types didn't include the Babylonians, so I can't speak to that particular aspect of your dad's beliefs.

He also says that they were interested in Buddhist mysticism, which is why they apparently chose the swastika, a Buddhist symbol, for their image.

The swastika was actually a very widely-used image in ancient times, used across the world in different contexts - sometimes as just a good luck symbol, sometimes as a sacred image, such as in Buddhism, Hinduism, and Jainism. Early archaeological discoveries of the swastika in Germany encouraged its adoption by nationalist groups (völkisch) as a symbol of national identity. Hitler claims in Mein Kampf that he set the final shape, color, and orientation of the symbol that became the Nazi standard. In this sense, the Nazi's use of the swastika might be considered coincidental.

He also said that they were trying to find some Tibetan monks' cave system where apparently they went deep into the earth and made an underground city structure where they became enlightened. And apparently the Nazis knew about this stuff.

Going back to the 1938 Tibet expedition, Tibet had been a major focus for several occult and fictional treatments in the 20s and 30s. The Theosophists, mentioned above, believed that the Secret Chiefs of the Great White Brotherhood had their hidden headquarters in an underground city in Tibet, for example. In 1871, Edward Bulwer Lytton published the novel Vril, the Power of the Coming Race, which was later linked to the Nazi expedition to Tibet, Theosophy, and occultism in the influential if spurious book The Morning of the Magicians.

Again, I am awful at history, but I'm skeptical. I'd love to know if any of this is true and to what extent, because he won't give me any sources for where he found this information (and believes that he shouldn't have to, so...).

There's a lot of material out there, a good bit of it of dubious scholarship. The idea of Nazis obsessed with the occult plays into a lot of aspersions denigrating the Nazis and their ideology, and has fueled hundreds of stories, novels, video and roleplaying games, comic books and manga, etc. It's a rich source of material for speculation that spurs a lot of imaginations and results in a lot of creative output.

But there is a bit of truth to the idea - just nowhere as extensive and straightforward as Nazi expeditions for lost Biblical relics. For example, just this year a portion of Heinrich Himmler's "witch library" was discovered.

Heya, you might be interested in our FAQ section on Nazis and the occult.

In particular: /u/commiespaceinvader provides a great overview with reading recommendations, /u/idjet talks more specifically about Nazism and the medieval occult, and /u/Bernardito looks at Atlantis in the Nazi imagination.

Since there's already a very good answer, I would just add an example.

Himmler seemed to be one of the nazis who was into the Occult. In 1940, just a year after the Spanish Civil War ended, with Nazi-friendly Franco in power, Himmler visited the Montserrat Abbey, in Northeastern Spain (near Barcelona.) He was looking for the Holy Grail there.

Apparently, there was some aura of mysticism around the mountain of Montserrat (where the abbey is) and it had been mentioned by romantics like Goethe. This article mentions that could be one of the reasons they might have looked there. They also say Wagner mentions the Pyrenees as the home of the Grail in his adaptation of the poem Parzival (which centers on the hero, Parzival, looking for the Grail), and apparently in Parzival they mention it is in Montsalvat, which sounds similar to Montserrat.

He didn't find it, though.

Here is another article in Spanish about some of the things the Nazis looked for in Spain (they mention the Spear of Destiny, among others), and they also mention the Tibet and South America briefly.

Finally, another Spanish article here about Himmler's visit to Spain in 1940 while Hitler was meeting Franco in Hendaye.