if you are coming to new york for one week you should buy a weekly train pass. you can use this on the train as well as the bus, so it is worth it. you cannot use the card between two people as there is a 20min gap between the first time the card is swiped before another entry is permitted.
The best and most economical way to travel around Brooklyn is by subway at $2 per ride, the one day Fun Pass for $7.50, 7 day unlimited $25, 14 day unlimited is $27 and the montly unlimited is $81. The subways runs 24 hrs a day/ 7 days a week but some lines in the smaller neighborhoods run less frequently.
The system has a labyrinth of networked connections which are quite easy to navigate. Check out the mta website for weekend schedules and interruptions and other fun ways to make the most of your trip.
Subway in Brooklyn runs above ground along McDonald and Brighton Beach avenues. It is a quick and easy means of getting to Manhattan. From there you can go to other boroughs, because all the lines loop through Manhattan.
Always make sure to check which trains are messed up. The weekends are especially insane. TIMEOUT New York has some insights, but it is safest to ask at the token booth for the most updated information. NY1 is pretty reliable, but Manhattan based.
Get a map at a booth and then ask ask ask and look for signs. Especially frustrating are the trains which stop on the weekends or the seemingly random express lines. Good luck!