Until 1937, Cuxhaven was part of Hamburg; in a reorganization two towns near Hamburg that were part of Prussia were given to Hamburg in exchange for Cuxhaven. With 55000 citizens, Cuxhaven isn't a very large city. It still can boast roughly 3 million booked nights a year, making it one of the largest seaside resorts in Germany.
Tourism is one of the traditional businesses of Cuxhaven. The other two are the fish industry, and the harbour. The city is trying to expand the latter and has invested a lot of money in the modernization and expansion of the port facility. Still, the economic situation isn't all too good and tourism remains the major industry of the city. Many people have left Cuxhaven in the past decades to settle elsewhere. Many of those who remain are retired, contributing to the small, quiet town ambiance.
Near Cuxhaven, between 1945 and 1964 about 500 rockets were launched . The best known rocket launches which took place in the area of Cuxhaven, were the launches of three V2 rockets in the operation Backfire, in order to demonstrate Allied forces the technique of the "wonder weapon" V2 in October 1945. The only thing remembering to rocketry at Cuxhaven is a trough near the way from Arensch to Sahlenburg.
Cuxhaven is a quiet place. If you're looking for action and parties, it's probably not the right destination. If you're looking for a quiet city by the sea, Cuxhaven becomes a much more interesting choice.
Most people come to Cuxhaven either via train or car. Some ferry services exist, but they are overall negligible.
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.
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.
There are limited choices when it comes to moving around Cuxhaven. You can walk, bring your own car, hire a taxi, or use one of the bus lines.
You can find a map of the bus lines here (pdf): 
The Nordersteinstraße is the primary shopping area of Cuxhaven. It is geared towards the general population and not really too touristy. Prices are rather normal as a result; the downside is that you cannot expect a store to accept credit cards. "EC" bank cards are usually accepted, however.
Cuxhaven is a traditional fishing town, so do not leave out some fresh fish or crabs while you are there.
When enjoying a walk in the Wadden Sea in front of the coast, you must take some simple precautions as the rising waters can be dangerous. If you are far from the beach and the tide surprises you, your life will be in danger.
Always remain within sight of the land, and do not walk out too far. Take care of fog and mist. Enter the area only after high and prior to low tide, and return shortly after low tide, i.e. as soon as the water starts to return and the high tide sets in. Keep a watch with accurate time, and memorize the times of low and high tide. The water does not rise in all places at the same speed. Some areas are lower, and Priele (creek-like arms or canals of water) can well flood very early, cutting you off from land.
You should ask for advice and rules on-location; there are several lifeguard stations and most tourist information places will be able to provide you with basic information as well. Ask for a tidal calendar (German: Tidenkalender).