I was watching a video about Jews in Israel with Iraqi ancestry, and I was surprised to find that many of these people have mixed ancestry.
For instance, one lady mentioned her grandmother was Ashkenazi Jewish German, but her grandfather was descended from Sephardic Jews in Iraq.
What was Israel society like in those early days? I'd assume there was some solidarity because of the same religion and rituals, but there probably was a lot of differences when it came to culture and certain customs.
When it came to social events, were all these Jewish groups fraternizing with each other?
Yes. This is still an issue in Israeli society today.
When the state was formed, the majority of the Jews in Israel were of European origin, because Political Zionism started in Central Europe and took the strongest hold in Eastern Europe. Therefore, basically all institutions and authorities in early Israel were presided over by secular, Ashkenazi Jews, who's culture, political ideology and mentality were those of early 20th century Central Europe. Meanwhile, a lot of immigrants were coming from the Middle East and North Africa, and they had to leave their homes in a hurry without many possessions or money, due to the backlash that the Arab-Isareli war created in the Muslim world.
So, one problem was the difference in culture between Ashkenazim and Mizrahim. The Ashkenazim generally looked down on certain aspects of Mizrahi culture (as well as traditional practices), and since they were in charge, they, if not tried to suppress it, at least didn't give it representation. For instance, until the 1980's, traditional Middle-Eastern music wasn't played on Israeli radio stations. It was still uncommon until the 2000's. Same is true for other aspects of Mizrahi culture. In schools, Mizrahi children were taught by Ashkenazi teachers, who didn't care about their cultural practices, and disciplined them to live the "Ashkenazi way of life".
Another issue, was that the young State of Israel didn't have many resources. When most Mizrahim came, without many possessions and money, the country was just recovering from the Independence War and still trying to figure things out. The poor immigrants were therefore housed in what basically amounted to refugee camps, which led to feelings of discrimination and class-difference among Mizrahim, who made the majority of poor immigrants. Those feelings have persisted to present day, though much diluted. Israel never had a Mizrahi prime minister, but the ruling party (Likkud) owes most of its support to Mizrahim, as it was the first large party to cater to Mizrahi population and promise better treatment to them, in the early 1980's.
That said, things weren't so clear cut, and there was definitely a lot of inter-mingling among groups, and Mizrahim were welcomed among Ashkenazim as long as they adhered to their way of life. The army was an important institution in that regard, because it forced all (well, most) different sectors of Israeli society to interact with each other as equals. But, mixing into a "melting pot" (which was a popular idea back in those days) is a very slow process that is still far from being completed. Cultural and ethnic differences have diminished over the years, but they are still one of the main topics of Israeli society today.
Source: am an Israeli who studied this topic to a certain degree.
While more can always be written, I have three answers on similar topics linked here to start off with.