First some sources and resources which are not exactly but quite near of what you want. If you would by chance be interested in early 17th century Portuguese Indiamen galleon / nau I would recommend this extensive PDF article: Outfitting the Pepper Wreck by Felipe Castro. Also a book I have not read (but is on my to read list is) which goes into a lot of details about daily life of Spanish transatlantic voyages of 16th century is Spain's men of the Sea. I think it is what you are looking for for info about amusement and entertainment. If you are willing to wait for oh, i hope a year, I will get around to reading it and could provide you more details

Now to some information. First the boring part.
If you were going to Goa it would be no sooner then 1510 when Goa was captured. Still until 1530s the Portuguese presence in India was headquartered in Cochin, so you would more probably be headed there. But the ocean winds would bring you closer to Goa first so it was usually a stop.

Trips to India were at the time organized as yearly Armadas (Carreira de India) so if your ship went to India it would not go alone but in a fleet. After the 1510s those fleet would number around 10 ships. Later (since 1530s) it would be lower then 7

The principal ship to go to India was the Carrack or as Portuguese called it the Nau. For reference it was a Santa Maria like ship (while Pinta and Nina were caravels), even though Santa Maria was a early smaller type of carrack. The carrack would have 3 or four decks and could hold around 100-150 sailors with additional 200-300 people (soldiers or slaves) and still have room for cargo. The sizes of ships were not standardized and varied greatly from ship to ship.

Caravels were also used in the early 16th century on the India run, as escort vehicles, scouts, fast ships and messengers. Also I found some theories that because caravels were lower (closer to waterline) then carracks, that made them more suitable for anti-ship cannons and would use their maneuverability to position themselves properly and destroy enemy ships. Upon arriving in India caravels would usually remain there for duties and not return back home. Additionaly when Goa shipyards opned they started making caravels (and carracks and galleys even!) built in India too.

Not much concrete data is available on caravels of that period. They had crew sizes around 20-30 and were quick and maneuverable. They could withstand the ocean and sail in (not to shallow) rivers. They had two or sometimes three decks and could carry around 100 slaves. They had two or three masts, and on the India route had square-rigged sails (for ocean sailing) with the aft sail being lateen sail for better maneuvering (for close to coast sailing). They had ability to change the sails configuration but had to stop and do it on land.

I don't know much about the daily life of sailors, especially on caravels. During the day sailors would do their tasks, but I would prefer not to try go into details, as I might embarrass myself as I don't really know that much about ships, ships terminology and ship tasks. Navigators and pilots would periodically through the day estimate positions, directions and speeds and note in the logs and charts.
Sailors would be sleeping on the main deck outside, or on the top inside deck, mixed with various animals brought on the trip. Where exactly would one sleep depended on the person's job, status and experience, with officers and passengers usually sleeping in cozier areas towards the stern.

Food was typical sailor food at the time. Biscuits (hardtack) given with water and wine, all carefully rationed. Salted meat and fish. Sometimes fresh fish if there was opportunity to catch it. Dried fruits and various nuts. Pickled stuff also. In the beginning of the journey fruits and vegetables, but as it is highly perishable it would quickly run out. Ships had it's own rations, but everyone was entitled to bring some personal belongings (how much based on rank) which was often extra food

As the outward trip to India involved a large amount of sailing far from land, getting fresh rations was impossible and sickness and scurvy were a serious problem. When I go through my sources I will provide some numbers on the losses of ship and men during journeys, but they were substantial.

Upon reaching harbour I suspect not all sailors could go to mainland, and I wouldn't really speculate what would sailors do. Probably stick to safe, Portuguese elements of Goa, and try to trade what little possession they had for items to bring back to Portugal to resell, which was common.

I know I couldn't really give much info about the daily life of sailors (except recommending a book about it) but I tried to tackle some of the questions

I'm a bit late, but I'd like to add to /u/terminus-trantor's response.

In the Portuguese armadas, as was mentioned, the caravel was actually not the most common type of ship because of it's low cargo capacity for supplies and merchandise. Caravels were mostly relegated to light escort and dispatch, while the most common ship was the sturdy nau.

In the 16th century, a voyage between Lisbon and Goa took about 6 whole months. Caravels could actually half that time at full speed, but it really was rare for a caravel to travel alone.

It's important to state this, in the Portuguese navy, there was a stark social difference between captains and lowly sailors. The captain of the ship was necessarily a man of noble birth, or fidalgo, who may by chance have some navigational skills (like Duarte Pacheco Pereira or Afonso de Albuquerque, but the crushing majority didn't). Then there was the pilot, then the master, then the sailors, cabin boys (aspiring sailors) and slaves. Roughly. Side-by-side travelled the gunners, soldiers, and assorted passengers, like merchants and clergymen, but the majority of the crew was usually soldiers or sailors.

The captains had their own cabin, and usually spent their time overlooking the ship as the head authority, but also reading, feasting and engaging in duels. The pilots would spend most of their time taking notes with charts and navigational books, though the rocking of the ship made accurate calculations basically impossible.

The masters overlooked the sailors. Now, to be entirely honest, after the ship's rigging is set and she is well on her way on the high seas, there's not really that much that needs to be done, other than tending to the rudder and keeping an eye on the weather. Sailors mostly found distraction in card games (the Portuguese spread playing cards across the world as far as Japan) though they were strictly forbidden to gamble to prevent brawls but, those found a way to happen anyway. They also sang, danced and of course, listened to mass at sundays. The clegymen would perform their religious duties on the high seas (though the drinking of wine was avoided to prevent the wine from spilling), and lead processions across the ship, up and down the decks. They entertained the crew with stories of saints and notable persons, and of course zealously kept an eye on 'sinful' acts.

A journey between Lisbon and Goa would pass by the deserted coast of Mauretania, the tropics and the arctic latitude by the Cape, before again feeling the warm embrace of the damp Indian monsoon weather. Of all these, the tropics were arguably the worse:

Under the unsufferable hot sun, shade was scarce and the ocean reflected the light; the dark hull of the ship turned the hold into an unbeareable greenhouse, and for that reason, sailors preferred to sleep outside. Note that this was also before the generalization of the typical ship hammock, so conditions were particularly cramped and uncofortable. The water boiled, fleas and bacteria proliferated and foodstuffs rotted. Understandably, many perished or despaired under these conditions.

Occasionally, some ships might call on the Brazillian coast, though this was discouraged since this entailed delays that might hold your jorney back for another entire year. Upon reaching the shore, some broke out the fishing nets and the arquebuses to hunt some game, either for sport or for some very welcome fresh meat. Fresh water was also brought in.

To make up for the exhausting heat, next up you'd face the strong cold gushes of the Arctic winds, meaning there was a lot to do around the rigging to keep the ship from capsizing outright, in soaking wet clothes with little possibility of warming up besides on any spirits (aguardente!) you might have brought along. Being a sailor in the Portuguese India Armadas in the 16th century was definitely not an enviable position. ...Many realized it then too.

My main source among others is actually a fascinating book about Portuguese medicine during the Discovery and Expansion Age that I just finished reading called História da Medicina Portuguesa Durante a Expansão (History of Portuguese Medicine during the Expansion) by Germano de Sousa, though sadly it's in Portuguese only I believe.