The word "Arab" has become synonymous with "Islam", though this is by no means wrong since the religion is a huge part of Arab identity, culture and history.
However, we never really get taught about the history of the Arabs before the Prophet Muhammad PBUH. I recall that during the period of war between the Byzantines and the Sassanid Persian Empire, the Byzantines enlisted the help of a kingdom of Christian Arabs, but that's just about all I know.
Could you give me recommendations and books to read about the pre-Islamic history of the Arabs?
The subject of Pre-Islamic Arabia is one that I have posted on before, and if anybody is interested in the comments I made at the time then here is the link to my prior commentary.
However, I'd rather not just post a link to my prior answer, especially because your question is not identical (though almost so) to the one I answered previously.
My book recommendation is Arabia and the Arabs: From the Bronze Age to the Coming of Islam by Robert Hoyland. The focus of the book is on exactly the territory that you're interested in, and the book is very much attempting to be comprehensive. However, it is also an area in which there are limitations to the evidence and gaps so do expect to be frustrated by the subject a little bit. Our understanding of Pre-Islamic Arabia as a whole is only improving over time, but the process is slow and the evidence tricky.
One thing I would say about Pre-Islamic Arabia is its boundaries do not necessarily follow those of the Arabian peninsula. Even prior to Islam there had been Arabic peoples who had settled areas outside of the peninsula. Likewise, one should not imagine the peninsula as being culturally homogenous, particularly prior to Islam- in Pre-Islamic Arabia attention is generally given to three or four regions/zones as identifiably distinct from each of the others. This is traditionally East Arabia, South Arabia, North Arabia, and I've noticed a number of historians of the era also distinguish the peninsula's interior from the other three.
For example, the Nabataeans were a people who at one point controlled territory ranging from the Sinai peninsula to nearby Damascus. They either adopted Aramaic but shifted to Arabic over time, or they used Aramaic for their writing system despite speaking Arabic (I am more convinced by the latter). They had quite intense interactions with Ptolemaic Egypt, Hasmonean Judaea, and the Roman Empire. They are also likely the middle men via which Aramaic script shifted into what we call Arabic script, and they may have spoken a direct relative to the modern Arabic language.
But by contrast, in Yemen, you have the Sabaeans- they used a different writing system, called the Old South Arabian alphabet conventionally. Their language, called Sabaic (but you may see incorrect references to Himyarite/Himyaritic, and if you do they mean this language), was part of a family of Arabic languages found in the South of Arabia and was not mutually intelligible with that of those in the North. Their main sphere of interaction was with other states/peoples in South Arabia, and the Red Sea; at various points, the Kingdom of Aksum in what is now Ethiopia/Eritrea actually conquered land in the South of Arabia and fought wars with the Sabaeans among others.
These are not the only two societies/states counted as part of Pre-Islamic Arabia, not by a long shot, but I used them as examples to illustrate the diversity of the peninsula. Among other things, many of these societies worshipped very different gods to one another, and also absorbed different practices from other peoples.
Before Islam most of the Arabian Peninsula were tribes that were fighting with each other. Southern Arabia, present day Yemen had been the starting point of the Frankincense trade which brought wealth to the region from their trading partners in Europe. This trade has helped build up cities like Mada'in Saleh in present day Saudi Arabia and Petra in present day Jordan. Excuse my lack of sources because I remember learning about this in highschool. Also you must note that when the message of Islam started, Muhammed Pbuh wanted to erase the old image of Arabia once he started to unite the tribes. Pre Islamic Arabia had many wars and tribal grudges (according to the poems that survive to this day).
Edit: this is not a full answer to your question but rather a snapshot of pre Islamic Arabia
It was my impression that there was no Arab identity per say at that point. Before the advent of Islam, "Arab" was a term used to identify only the Bedouins. Actually, if you read the Quran, it never even mentions the word Arab.
Source: The Arabic Language by Kees Versteegh and C. H. M. Versteegh
However, there seems to be more history and a longer history of the Afro-Semitic Language Family.
This ASLF goes back at least 8,000-10,000 years ago, and its roots are in Ehtiopia. It includes the speakers of Hebrew, Aramaic, Arabic, Amharic, Babylonian, Old Egyptian.
That these languages all came from a common proto-language spoken ~10,000 years ago in Ethiopia - I personally think this is amazing.