To avoid undue generalisation (and in part due to limited knowledge), my answer will be based purely upon two sets of Qing Dynasty sons – those of the Kangxi (1654-1722) and Daoguang (1782-1850) Emperors.
To begin with, it must be noted that the method of succession under the Qing was not based upon primogeniture – a departure from the Ming. Instead, the Qing emperors were nominally to select one of their sons to be heir. For example, the Daoguang Emperor had to decide between his eldest (surviving) son, Yizhu (1831-61), and third son, Yixin (1833-98) to appoint as his heir, and chose Yizhu over Yixin, apparently due to his prioritising filial piety over administrative competence when the two asked what their first act as emperors would be.^3 The problem of selecting an heir could thus become quite contentious, especially if no successor had yet been appointed.
The succession to Kangxi in particular was rife with dynastic politicking. His second surviving son, Yinreng (1674-1725), had been personally groomed from birth to become Crown Prince, but the discovery of severe sexual depravity (both real and alleged) led to his disinheritance in 1707. His elder brother, Yinzhi (1672-1735), tried to secure his execution and the appointment of Yinsi (1681-1726), the eighth son, as heir, but was himself stripped of title in retaliation. In 1709, believing Yinreng to simply have been suffering from a bout of mental illness, Kangxi restored his title as Crown Prince, only to revoke it in 1712 when he attempted to stage a coup whilst Kangxi was outside the capital.^1 In the event, the succession of fourth son Yinzhen (1678-1735) as the Yongzheng Emperor was incredibly insecure – factions supporting the 13th and 14th sons had been established years before – and his legitimacy as ruler consistently challenged.^1 ^2
The problem of disinheritance was also particularly severe under Kangxi. The conspiracy against Yinreng involved allegations about the use of sorcery by Yinzhi and Yinsi (1681-1726), with Yinzhi apparently attempting to curse Yinreng through Tibetan lamas and Yinsi being supported by a fortune-teller (who was executed by Kangxi to prevent undue attention being granted to the succession problem).^1 Even after one's brother became Emperor, one could still get in severe trouble, as Yinsi, who had been in a senior administrative role, ended up not just disinherited but expelled from the Aisin Gioro clan during Yongzheng's reign.
As a Qing Dynasty prince, one could expect to be granted a post of civil or military authority (of course, assuming that one was not disowned), but by specific appointment rather than by inheritance. When Kangxi went to war with the Dzungar Khanate in 1687-97, Yinreng remained in Beijing as regent, but many of his brothers, including Yinzhi and Yinzhen, held military commands (albeit subordinate to more experienced Manchu generals).^1 Even as brothers to the reigning emperor (or indeed uncles), princes could be given heavy responsibilities, Prince Gong (a.k.a. Yixin, son of the Daoguang Emperor) becoming Prince Regent in 1861 upon the death of his brother, the Xianfeng Emperor, and establishing the Zongli Yamen – China's first unitary foreign ministry – the same year. As Prince Regent until 1865, Prince Gong presided over the effective defeat of the Taiping Rebellion (1851-66) with the recapture of Nanjing in 1864, having been partially responsible for securing Western military support for the dynasty, as well as seeing the Qing through the height of the Nian Rebellion (1851-68).^3 However, one was not safe from dynastic politics simply by being the reigning emperor's brother rather than son – As noted earlier, Yinsi was one of Yongzheng's senior bureaucrats and effectively a cabinet member at the time of his downfall in 1725.
In summary, the most important aspect of Qing princes' lives (albeit chiefly when there were multiple sons) was dynastic politics – not just (indeed in most cases not even) in securing the succession but also staying on the Emperor's good side to prevent a fall in rank. Moderate rank could be attained as the emperor's son and even higher posts as his brother or uncle, but the threat of deposition continued to loom.
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It was highly dependent on the dynasty, and even emperor in question. Though it was considered "moral" and "proper" for a father to designate his son as his heir, there was actually no law stipulating that it must be so. The emperor, in theory, had the absolute prerogative to chose whomever he wished from among the royal family as his successor - or like as not, free to simply choose no successor whatsoever (and damn the inevitable chaos that was to follow).
As a particular example - since I've been looking closely at the formation of the Song Dynasty as of late - I'll give you the curious case of first two Song emperors: Taizu and Taizong.
Song Taizu (né Zhao Kuangyin, 927-976), the dynastic founder who had in 959 pulled a fantastic judo-maneuver with the Northern Zhou army he commanded and turned it around to march back on his own capital, thus usurping the throne, had no shortage of sons. In fact he had five of them, altogether, the eldest of which was named Zhao Dezhao. Dezhao was born in 951, meaning that he reached the “typical” Chinese age of majority between 966-68 (it tended to be a variable age somewhere between 15-18 depending on the individual and circumstance… though it could sometimes range significantly higher, into a man’s 20s).
In any case, Taizu was one of those Chinese emperors who just … never got around to officially proclaiming an heir. And the strangeness went even beyond that – over the entire course of his 16 year reign, he’d only ever name one individual as a 王, or “prince”, which was of course a necessary first step in legitimating a succession order. But even though his eldest son had turned 18 in 968 – the age at which he was expected to have been ennobled and given title – it was never forthcoming. Instead, it wouldn’t be until 973 that he would at last receive formal rank and title – but still not as a “prince of the blood”. Instead, poor Dezhao was granted the rank of jiedushi, or “military-governor”… a position that once upon a time during the glory days of the Great Tang Dynasty, been a truly formidable post, but had by now (and very much by Taizu’s own doing) been enfeebled into little more than a ceremonial position. Dezhao was also granted another ceremonial title which on paper made him equivalent to a chancellor of the court, but again had little if any real authority.
The one and only prince created by Emperor Taizu was instead his own younger brother, Zhao Guangyi (né Zhao Kuangyi)… who had been thus ennobled in the same ceremony as Zhao Dezhao, apparently just as one further slap-in-the-face to his own son. Guangyi – now Prince Guangyi – was already the 2nd most powerful man in the empire, serving as the governor of the capital city, Kaifeng. And his 973 ennoblement only seemed to confirm what many already assumed: that he was the heir apparent to the Song Empire, not one of Taizu’s sons. This would bear out in 976 when Taizu died (under rather questionable circumstances). A tale woven by Sima Guang in the 11th century historical compilation, Zizhi Tongjian, spins a tale in which upon being informed that her husband had died, Taizu’s widow, the Empress-Dowager Song ordered her eunuch servant to “fetch the Crown Prince” for immediate accession as the new emperor. She of course meant her own son, Dezhao. But the eunuch understood (somehow) that “Taizu’s intention to pass the throne on to the Prince of Qin [Zhao Guangyi] was widely known”… instead went and fetched him to the palace. Upon entering the throne room, Empress Song was visibly shaken, and said nothing but “the lives of myself and my son now rest with you.” It should be noted that, though this whole story is nice and juicy, it’s very probably fictional. There are more holes in Sima’s telling of these events than a block of swiss cheese – some which directly contradict what would have even been feasible at this time and place. Modern historians Lau Nap-Yin and Huang K’uang-Chung write that especially given his singular ennoblement in 973, “it seems clear that by that time Guangyi was already Taizu’s intended successor.”
As for poor, snubbed Zhao Dezhao? Well, he wouldn’t have too terribly long to mope. In spite of Guangyi’s (now Emperor Taizong of Song’s) apparent assurance to his sister-in-law that she and her family would be perfectly safe under his care… circumstance changed rather rapidly for the young Song Dynasty. Just before dying, Taizu had breached an nonaggression-pact with the powerful Khitan Liao Empire to the north in the hopes of completing his reunification of China by destroying a Liao vassal state called Northern Han. That hadn’t panned out, but Song Taizong had no intention to let the matter drop. In 978, he did manage to steamroll N. Han, but then got greedy and decided he wanted to “retake” 16 prefectures along the Liao-Song border that had been ceded some half-century prior. His army besieged the Liao southern capital city of Yan [mod. Beijing] for three months, before Khitan mounted reinforcements arrived, trapped, and then crushed the Song infantry. In the aftermath of this, grumbles began sweeping through the military that maybe Taizong wasn’t their guy after all, and maybe they should encourage Zhao Dezhao to take the throne for himself. Now, a military coup was precisely how the Song had come into existence, so as you may well imagine, its leaders were rather particularly sensistive to any such mutinous whispers (indeed, that’s why Taizu had so thoroughly de-powered the position of Military-Governor). And so Taizong did what he felt he had to do: he ordered his nephew Zhao Dezhao to commit suicide.
Lau, Nap-Yin and Huang K’uan-chung. (2009) “Founding and Consolidation of the Sung Dynasty under T’ai-tsu (960–976), T’ai-tsung (976–997), and Chen-tsung (997–1022)” in The Cambridge History of China, vol. 5 Part 1: The Sung Dynasty and Its Precursors, 907-1279.