The following taken from Wikipedia

The Mayflower departed Plymouth, England, on September 6, 1620, with 102 passengers and about 30 crew members, 74 men and 28 women, in the 106 feet (32 m) long ship. The seas were not severe during the first month on the Atlantic, but in the second month the ship was badly shaken by strong north-Atlantic winter gales, causing leaks from structural damage. There were many hardships and dangers throughout the trip, including seasickness and the bending and cracking of a main beam of the ship. One death occurred, that of William Button

After two months at sea, they sighted land on November 9, 1620, off the coast of Cape Cod. They attempted to sail south to the designated landing site at the mouth of the Hudson but ran into trouble at Pollock Rip, a shallow area of shoals between Cape Cod and Nantucket Island. With winter approaching and provisions running dangerously low, the passengers decided to return north to Cape Cod Bay and abandon their original landing plans.

The Mayflower dropped anchor in Plymouth Harbor on December 16 and spent three days looking for a settlement site. They rejected several sites, including one on Clark's Island and another at the mouth of the Jones River, in favor of the site of a recently abandoned settlement which had been occupied by the Patuxet tribe. The location was chosen largely for its defensive position. The settlement would be centered on two hills: Cole's Hill, where the village would be built, and Fort Hill, where a defensive cannon would be stationed. Also important in choosing the site was the fact that the prior villagers had cleared much of the land, making agriculture relatively easy. Fresh water for the colony was provided by Town Brook and Billington Sea. There are no contemporaneous accounts to verify the legend, but Plymouth Rock is often hailed as the point where the colonists first set foot on their new homeland.

The area where the colonists settled had been identified as "New Plymouth" in maps which John Smith published in 1614. The colonists elected to retain the name for their own settlement, in honor of their final point of departure from Plymouth, Devon