The Mayflower took 9 weeks to travel from England to the New World. The ship didn't land at Plymouth Rock until later, as the ship anchored in Cape Cod for a time while a small group scouted the area.
The Mayflower departed from England on September 6, 1620, after several tries; originally, the ship Speedwell was also supposed to make the journey, but as it had leaks, it was decided that the Mayflower would make the trip alone.
The ship reached land on November 9, 1620. The pilgrims didn't decide on Plymouth Harbor until after an exploring party had scouted the area. In fact, it took them over a month to decide on Plymouth as the place where they would build their settlement. A large part of the group lived on the ship until March of the following year, though they would disembark to find food and scout the area.