The voyage that took the Portuguese explorer across the Indian Ocean was a turning point in European history as well the History of colonialism. In this context
To start we should mention that it was in fact Bartolomeu Dias that first navigated across the Cape of Good Hope in 1488, but stopped a little bit further and returned as per his crew request. I am not mentioning this to correct a false premise, but because this is important as the experiences of Dias' voyage - and the preceding voyages by Diogo Cão, as shown in this map- made Portuguese realize that coasting down the Africa coast was too long and exhausting, and that the best course of action would be instead to turn south west after passing the equator and then when reaching the latitudes where you could find prevailing westerlies winds, to turn east and be propelled pass the Cape of Good hope (here is a map of an example route). This mandated some changes regarding choices of ships and ship preparation. Namely, the lateen sailed caravels were to be replaced by newly built carrack type square rigged vessels that could carry more supplies and be more suited for ocean travel.
But the first thing to note is that from different sources we have somewhat conflicting information about the ships, and no exact details about the ships sizes, crew count or mismatch in their types and names, although the differences are minor. The consensus currently is that Da Gama fleets consisted (at the start) of four ships, two carrack type ships of tonnage of around 100 tons (this is basically a small large ship) named São Raphael and São Gabriel, a caravel of 50-80 tons size called Berrio or alternatively São Miguel, and a unnamed store ship with skeleton crew that was used only on the first leg of the voyage to carry supplies, and was dismantled as planned before proceeding through the Indian ocean. To proceed with your questions:
I mentioned that the two carracks were newly built from wood specially cut for this purpose. They were also specially supplied for the voyage, with extra sails, tacks, rigging, equipment and supplies. But it's really hard to say if they were of any kind of new design, and most likely they weren't. They were the typical three masted carracks (or less likely four masted), with square sails on the front two masts, and the lateen sail on them mizzen ones. This design had a good trade-off of being a good ocean sailer, as well as keeping somewhat the ability to tack and sail close to shore. However the design is hardly new, as it existed for at least 50-100 years previously. A new thing used might be the latitude navigation by measuring the altitude of celestial objects. Measuring the altitude of Polar Star was introduced since the 1460s in Christian Europe (navigators in the Indian ocean used it, with some different tools for some time previously) and a new navigation by measuring the altitude of the Sun was introduced by the Portuguese in 1480s. It certainly was used by Vasco da Gama, but the full benefit would be reaped only by the following expeditions as knowing your latitude means nothing until you know the latitude of your destination. Which Da Gama had no clue. My old posts on Portuguse carracks and caravels of the period
The ships crew count is also one of the unknowns. The total expedition crew size given in source range from 120 to 170 (with some unrealistically going to 260). Guesses on their spread over ships are usually given as 70 on the flagship, 50 on the other carrack, 30 on the caravel, and 20 on the store ship (for the 170 count). Some 55 were said to have died during the whole expedition. For the quarters, officers had their cabins in the sterncastle, with the crew in turn sleeping in the forecastle, or on the deck outside if the weather permitted, and was preferred than the lower deck (which would be used in bad weather) which could be too hot, smelly and lacked proper air ventilation. Overall, the conditions were horrible, very unhygienic, and you could forget any wish for privacy. Supplies were stored in the hold of the ship (the lowest part of the ship), mostly in barrels, but also casks, jars, sacks, boxes were used. The common rations would be supplied by the officer in charge. Each person on board could bring some of his own stuff on board and store it in his personal chest (well, semi-personal. they were often shared). Many brought food as their private stash. Food would be cooked independently by the sailors on deck, and often few sailors would gang together and cook jointly. In few of my old answers I talk more about it.
Sanitary conditions were abysmal. It seems the most common method was to use chamber pots and then throw the waste overboard. Some textual sources of Iberian ships talk with much horror and embarrassment of alternate option of basically hanging from the side of the ship, holding yourself by the ropes, with a high chance of falling overboard. For later ships we know they had designated facilities at the bows were refuse would through a tube fall to the sea, but 1500s had sterncastles and no such bows. It is possible as some contemporary illustrations suggest that parts of the ship that protruded over the hull (like the parts of the fore- and stern- castles, barrels, turrets, outhouse like structures) would have holes , and as such were likely to be used as toilets, but nothing is conclusive. Book Those Vulgar Tubes by Joe J. Simmons is a good overview of sanitation of ships in the period up to 1800
Archeological remains of a Portuguese vessel in the Indian ocean of 1502, show presence of different cannon balls, some stone (24- or 32-pound stone), fewer cast iron (~6 or ~12-pounders) and a lot of composite iron-lead balls of smaller calibre used in numerous swivel cannons that seemed to be the favorite of portuguese as they lined the decks in great numbers. From personal weapons we found arquebus barrels of small calibre. From documents of other expeditions we know Portuguese also carried crossbows, spears&pikes, swords etc. From armour breastplates and helmets. In this old post I talked a bit more about weapons and artillery