So, for the era of 1480-1520 which encompasses famous expeditions of Dias, Columbus, Da Gama, Magellan etc. the standard operating procedure was to anchor close to shore in a place suitable for anchoring ( a bay secured from winds or other natural harbors ) and then use smaller boats and launches to reach the shore.

Only a part of the crew would go as some men, or more like most men, would stay on the ship(s). The most famous example is Columbus initial landing in his first voyage, which he colorfully describes here, where he and rest of his captains, flying their banners high reached the shore.

Another example can be found on Da Gama's expedition (you can read it here) when upon reaching bay of St. Helen in modern South Africa, Da Gama himself with his captains and armed men (infantry and some crossbowmen) landed to trade with the natives. however it's worth noting that on other occasions, Da Gama wasn't among the first ashore, especially after few unfriendly incidents when he decided it's best for him to stay as close to his ships as possible and not allow himself to be placed in danger of being captured. When ashore in populated lands, Portuguese rarely ventured much inland. They mostly stuck close the their boats to get water, wood and food, and only sent smaller number of men to visit nearby villages, if at all.

From the crew that landed, if a party was to be further sent inland, a portion of the men would stay with the beached boats to guard them, or the boats would return to the ships - but this would be hazardous for the landed crew so the former option was preferred. An example of such an event comes from Magellan's expedition, although the instance is not an usual one. It involves a war expedition where Magellan himself died. You can read the description here and basically from 60 men Magellan took with him, 49 were to go ashore and fight while 11 would defend the smaller boats. The rest were left on board ships. The expedition was a catastrophic failure and the catalyst of several events that brought demise to a number of people of the mission.

Finally your questions ask if the crews ever beached their ships. This would be rare, but indeed it did happen in special circumstances. Namely, in Da Gama's expedition, after concluding his stay at Calicut Portuguese stopped at Anjediva islands to repair and refit his ships for the return leg. This involved, according to official journal (here):

On the beach, in front of the church, we careened the Berrio and the ship of the captain-major. The Raphael was not drawn up on the beach, on account of difficulties which will be referred to further on.

The beaching was done for a specific purpose: to careen the ship to fix any problems it had in it's hull. Careening could actually be done on the sea without beaching the ship, and was indeed the main practice with larger ships who couldn't really be beached. Yet Da Gama's expedition involved smaller ships (caravel and two smaller naus) which were obviously possible to beach and was done. I do have to mention that following this example we don't see much talks about beaching the ships. We do have (numerous) references to careening ships, but as said this could be done without beaching and we can only guess if the author meant careening on land or on sea.

You may also be interested in my previous answer to semi-related questions:

Where did sailors get fresh water from back in the day?

When sailors during the age of exploration went to foreign ports, how did they buy stuff personally? Did they have time off the ship to shop and buy stuff from merchants and if so, with what kind of money?