Would it be rare for younger nobles to search for personal wealth in the First Crusade or would that be reserved for later crusades- and if so, which crusade had the largest amount of individuals seeking personal advancement? How common was death during the First Crusade on the journey to Jerusalem?
The short version is that you probably don't join a crusade. The idea of crusades as a method of gaining wealth has long been a popular one, particularly in anti-catholic histories of the early modern era that wanted to paint the Middle Ages as irreligious and backwards. Downplaying spiritual motivations and promoting material ones was a big part of this, and it can be seen as still very influential in 19th century and early 20th century histories.
There certainly were nobles who joined crusades (particularly the First, but also some of the others) to gain power and wealth. Bohemond is the most famous of these, a Italian-Norman prince who had given up his inheritance in hopes of carving territory for himself from the Byzantine Empire but failed, and later joined the Crusade and created the principality of Antioch for himself. We can't know Bohemond's motivations for certain, but his actions do suggest a material motive. Baldwin (future King Baldwin I of Jerusalem) and Tancred (Bohemond's nephew) are also cited as likely candidates for material gain, Baldwin took the Principality of Edessa during the First Crusade, and while less successful (in part due to Baldwin's actions undermining him) Tancred also made some moves to carve out territory for himself. Godfrey of Bouillon (Baldwin's older brother) is more difficult to place, he eventually ended up ruling Jerusalem for a year before his abrupt death, but it is not clear if he intended to remain in the Holy Land when he left Germany. Similarly, Raymond St. Gilles (one of the crusade's most influential and wealthiest leaders) ended up settling the Principality of Tripoli, and may have attended to die in the Holy Land (he was no spring chicken when he left), but given his starting wealth it is hard to say that he intended to gain money by his expedition.
These careers suggest that the idea that some nobles might have used the Crusade to enrich themselves isn't without merit, but it is also worth distinguishing between people who achieved wealth as part of a Crusade and those who sought it as their reason for participating. Jonathan Riley-Smith was integral to upsetting the old image of crusaders searching for wealth by conducting several in depth studies that showed both that (a) most crusaders were not younger sons of nobles, but in fact wealthy nobles in their own right and (b) that crusading was insanely expensive. The cost of joining a crusade (which would include recruiting a retinue, paying for supplies both at the start and along the journey, and in the more nebulous cost of loss of revenue due to your absence from your lands) was astronomical, and in later crusades we see quite a few examples of people taking on substantial debts to participate (the cost of Richard I's participation in the Third Crusade is often highlighted as a particularly outrageous example). As a money making enterprise, joining a crusade was a terrible idea. On top of that, relatively few nobles settled permanently in the Holy Land. A lack of permanent residents from the west was a constant threat to the survival of the Crusader States, and was one of the most significant factors in their eventual collapse.
Should you wish to join a crusade anyway, and you were a lesser noble looking for a new life in the Holy Land, your best bet was to join the retinue of a greater noble planning a crusade. As a member of a retinue (ideally royal, but a duke or very wealthy count would also do) you could expect your lord to subsidise or possibly fully pay for your expenses of travelling to the Holy Land. Upon arriving there and fulfilling your obligation to your lord (or if you were very crude, by abandoning him, but this would not help your reputation and is not advised) you could seek sponsorship from a local noble, and maybe get yourself a nice little fief or subsidised income there. You would be very unlikely to get substantial plunder from a great victory or other glorious action, so I wouldn't hold out hope for making your fortune off of ransoming a captured Arab prince or anything like that. You are also unlikely to climb the social ranks in this action, some nobles did rise (Baldwin didn't start a King after all), but it was mostly important nobles becoming more important. The internal politics of the Holy Land were different from Western Europe, and you would have different opportunities, but the Holy Land was no realm of reinvention where you could make yourself anew. A Horatio Alger novel it was not, basically.
As to your likelihood surviving the First Crusade, it wasn't great. I don't have my books on me right now (trapped in work), but the short version is that a lot of people died on the Crusade. You would have had to survive multiple battles, one of the most gruelling sieges of the entire Middle Ages at Antioch, multiple threats of starvation, and eventually survive the final conquest of Jerusalem. As a noble your odds were better than that of a peasant, you were more valuable to other nobles so you could try and join the retinue of one of the Crusade's great leaders and they would then subsidise you with funds for food and other necessities to remain alive, but compared to your average Medieval military campaign the First Crusade was particularly harsh and difficult. You're marching thousands of miles through enemy territory, sometimes in blistering heat, with no idea what the terrain is like or where to find food, the threats to your life are substantial.
For further reading on this kind of topic Jonathan Riley-Smith's work is probably still the go to place. He wrote a short history of the crusades as a whole, as well as a history of the first crusade, and a whole swath of articles and specialist texts.
Christopher Tyerman's How to Plan a Crusade is a good study of all aspects of Crusade organisation and execution. Be warned, it's not a very engaging cover to cover read, but if you want to look up the answer to a specific question (or more information on a specific area) it's a great reference work.
Take a look at these previous answers by /u/eighthgear and /u/TheGreenReaper7: