The city center is easily explored on foot. Most places of interest can be reached walking. The downtown shopping district is a car-free pedestrian zone. In case you need to visit places further out, you can rely on the public transportation system .
By Public Transportation
The Braunschweiger Verkehrs-AG serve the city and the district of Braunschweig. Places within city limits are easily reached by public transportation, either bus or tram. Ticket prices vary depending where you need to go. The various zones of the Brunswick district can be found here.
Prices are available here. Tickets can be purchased at the driver, at certain stores, or via cell phone. The current price for a one-way ticket within city limits is 1,90 Euros.
Information on schedules and connections can be found either at the BSVAG itself, or on the EFA site, where you can look up connections for all of Lower Saxony and Bremen.
By taxi
Taxis are comparatively expensive in Germany, and Brunswick is no exception. They are usually used by locals on weekends after buses and trains stop running for a ride home, or if you need to move lots of luggage to, e.g. the train station. The concept of sharing a cab with strangers is foreign to locals and drivers alike, although some people can be persuaded to share a cab, if your stop is on the same way and when cabs are sparse e.g. in inclement weather.
Publicly registered taxicabs - those with yellow signs on top which say Taxi and are usually taxi-colored creamy eggshell color - can be phoned 0531 - 5 55 55 or 0531 - 6 66 66 or hailed. Private companies, such as MiniCar (http://www.minicar-brauns...) or CityCar (http://www.city-car-bs.de/), which are usually a bit cheaper only pick you up with a prior reservation.
By car
All major car rental companies have branches in Brunswick. But since Brunswick's highways tends to get gridlocked during rush hours neighboring Peine is the county with the largest number of commuters in Germany, most of which commute into either Braunschweig or Hanover. Moreover, Brunswick is rumored to have the highest per-capita-density of trafic lights in Germany. Try to count your number of stops when moving around by car in Brunswick.
Brunswick has many parking garages, which are organized through a pretty efficient parking guidance system. You can access information on spaces available, fees, and opening hours via your cell phone at wap.braunschweig.de