The Apollo 11 astronauts on the famed 1969 mission took 3 days, 3 hours and 49 minutes to go from launch to close lunar orbit, typical of manned missions. Unmanned spacecraft employ a much wider range of travel times.
The shortest voyage of a spacecraft from Earth to the moon was the Luna I probe launched by the USSR in 1959. It reached Earth's natural satellite in approximately 36 hours. Luna I was a flyby mission only, and though it passed close to the moon, it did not enter its orbit. The longest trip to the moon took more than 13 months, accomplished by the European Space Agency's SMART-1 lunar probe in 2003, using a slow but extremely fuel efficient ion engine.
1. Download a free moon map. These are easy to find, but the ARVAL Observatory in Argentina provides one of the best. 2. Pick a night either a couple days before or a couple days