To be fair, I have to point out that "options" are not really on the table, because as a royal son or daughter, you have duties to the realm that mean you can't necessarily just decide on a career of some kind. But the lack of "options" is true for most people of the day. Our ideas of finding the happiest possible way to live our lives would have been fairly alien to them.

Since you're asking about England in the 14th century, we might as well examine the children of King Edward III and Philippa of Hainault, who lived in that time. The couple had a number of children who survived long enough for this to be an issue: Edward (1330–1376), Isabella (1332-1382), Joan (1334-1348), Lionel (1338-1368), John (1340-1399), Edmund (1341-1402), Mary (1344-1362), Margaret (1346-1361), and Thomas (1355-1397). (There were also three who died very young: William (1336-1337), Blanche (1342), and William (1348).)

Edward, known as the Black Prince, was in line to be the next king, so he's beside the point. (Although it's worth noting that Edward III created the first English dukedom for him.) But he died before his father and so his son, Richard, became king after Edward III.

All of Edward's and Philippa's daughters were raised to be married. Potentially, they had the option to be placed in a nunnery, and noble parents often sent a young daughter to the church, which would give them Christian cred and, frequently in the long term, a familial relation to an abbess with money and land at her disposal. However, in a royal family, this was not as useful and not as common by the High Middle Ages. Contrary to the general opinion, daughters were considered a valuable resource for engaging in marital alliances, which would forge or strengthen links to other royal families or to important nobles in one's own domain. Isabella was betrothed several times, but she ended up marrying Enguerrand de Coucy, a French nobleman who'd been involved in the wars between the English and the French; he was supposedly her choice, and while it's a rather unimpressive marriage for a king's first daughter, she was actually superannuated for an unmarried princess - in her thirties. Joan was betrothed to Pedro, son and heir of Alfonso XI of Castile, but died of plague on the way to marry him - she would have become a queen eventually. Mary married John de Montfort, who had a right to be Duke of Brittany (which was, at the time, a relatively autonomous region), so again, she would have been a ruler's consort if she hadn't died young. And lastly (although not chronologically lastly - that was Isabella), Margaret married John Hastings, future Earl of Pembroke and part of the network of kin and friends surrounding the royal family.

The male children of Edward and Philippa, however, were all eventually granted titles of their own - royal dukedoms that would become traditional titles given to princes. Lionel was duke of Clarence, John was duke of Lancaster, Edmund was duke of York, and Thomas was duke of Gloucester. These titles related directly, at the time, to lands and estates granted and won through marriage, and the royal dukes had corresponding incomes and responsibilities. However, they were still their parents' marriage pawns, though entirely domestically; to some extent, they had to be, because Edward had given his oldest son a massive amount of land - roughly a quarter of the kingdom - and had relatively little left to give the others. Lionel was betrothed as a very young child to a slightly older heiress, who bore the title of countess of Ulster in her own right, and therefore made him earl of Ulster as well as duke of Clarence. John was married to one of a pair of heiress sisters (the other of which died, leaving him and his wife the sole beneficiaries of her £8,500/year income). Edmund was largely funded by his father by being given parts of lands of deceased nobles; he married the daughter of Joan's intended husband at the very end of his father's reign, and was granted his dukedom by Richard II. Thomas was also married to an orphaned heiress, although unlike John, he wasn't able to gain her sister's inheritance - because John married his son to the sister! (What a messed up family tree. That's what happens when your main idea is to grab up any available heiress.) Thomas also didn't get his earldom until the end of Edward III's reign, and wasn't enduked until Richard's.

Edward III granted these dukedoms as and when he did for a purpose: it signaled that the sons given titles were ready to take their place on the political stage. (Edmund and Thomas were likely not considered to have much authority among the English political elite.) His sons were very loyal, and therefore quite helpful to him as the earlier generation of nobles with personal loyalty and friendship for Edward were dying off. There were not many options outside of this - certainly becoming a monk or philosopher or historian and ignoring the outside world would have been out of the question. Just as the daughters pulled their weight by marrying and conducting diplomacy between their new families and their old one, the sons pulled their weight by marrying heiresses to support themselves and supporting their father's political goals. They also participated militarily: Prince Edward lead the English in a number of battles and negotiated treaties, and John was also quite active in the Hundred Years War; Lionel was made governor of Ireland and focused his efforts there. Edmund was also a military commander at the very end of his father's reign, and Thomas assisted his brother-in-law's efforts to regain Brittany. The royal dukes would also come to support (at least to some extent) their nephew when Richard II came to the throne in a very similar way. Royalty was a family business.

Here is a much earlier answer by u/knight117 to tide you over until someone else with a new, different, or longer perspective comes along.

https://www.reddit.com/r/AskHistorians/comments/2a0xxn/what_would_the_second_third_and_fourth_sons_of/