Most people come to Cuxhaven either via train or car. Some ferry services exist, but they are overall negligible.
By train
A real alternative to the car is a train drive to Cuxhaven. It will take roughly the same time, but it's more relaxing than using a car. However, the German rail company tends to be more expensive if you travel with several people unless you catch a special deal.Travelling with Happy Weekend Ticket or Länder Ticket allowing 5 people at a total price 37 Euros is a recommended option when travelling from Hamburg, since the ordinary train takes only around 1 hr 45 minutes for this journey.
Cuxhaven's railway station is right in the center of the city. The bus terminal is in front, and taxis can easily be hired as well. On foot, it should take you about five minutes to get to the Nordersteinstrasse shopping area.
By car
Major road not Autobahnen/highways lead to Cuxhaven. From Hamburg, it's about two hours along the B73; from Bremen the drive should take closer to one hour. Be wary that the roads also serve commuters to many smaller cities and villages in between and can be crowded and tricky. The B73 especially is known for motorists with bad driving style and deadly accidents; some extra caution should be employed.