A follow up on themicah's public trasportation advice:

1. The 5 train only runs to Brooklyn during weekday rush hours. If you're on a 5 train and it terminates at Bowling Green (the last stop in Manhattan), simply wait on the same platform for a 4 train.

2. I would change from the 6 train to the 4/5 train at 42 St/Grand Central because it's a simple cross-platform transfer (you would have to negotiate stairs/escalators at either 86th or 59th).

If you're doing this on a weekend:

1. The 4/5 (express) trains are running local on weekends (see here). If you're starting at 103 St/Lex Av, you can get on any train (4/5/6).

2. The 4 train does not run between Manahattan and Brooklyn (see here), so you would need to find another subway route. The easiest would probably be to switch to the R train at 59 St and take it to Court St in Brooklyn. This transfer involves stairs.

If I had luggage, I would use cab.