A nice way to visit Brooklyn is to take the subway and get off the metro station "York Street subway". The station is located on Dumbo neighbourhood. A couple of streets away from the station you will find a very well- known spot between water street and washington street, which has a great view to the manhattan bridge (and has many many tourists taking photos there).
Keep on walking along water st to brooklyn bridge direction and you will find a great view of the city and the bridge.
Don't get anxious and go to the Brooklyn bridge, before doing it you must visit another neighbourhood called "Brooklyn heights". If you keep on walking and pass through the bridge you will find another amazing view of the city. You will spend a lot of time here watching the city. Near this park you will find Brooklyn heights. This neighborhood has a lot of brownstone rowhouses, most of them built prior to the Civil War.
Once the tour finished round this great neighborhood we went to cross the brooklyn bridge. Usually this bridge is full of tourists and you should be very careful with the bike lane! Crossing/going over the bridge could take around 20 minutes or more since you will stop at many spots. The bridge is 1825 meters long and it was built between 1870 and 1883. The bridge carries around 150000 vehicles and pedestrians every day.
Nice facts about the bridge:
On May 17, 1884, P. T. Barnum led 21 elephants over the Brooklyn Bridge to prove that it was stable.