How do you get from Grand Army Plaza Brooklyn to Grand Central Station?

That depends on where in Brooklyn you're going (see the Related Link below for a complete New York City subway map).

The subway lines that run between Manhattan and Brooklyn are the 4 and 5 (of the 4-5-6, the dark green line), the A and C (of the A-C-E, the blue line), the B, D and F (of the B-D-F-M, the orange line), the N, R and Q (of the N-R-Q-W, the yellow line), the 2 and 3 (of the 1-2-3, the red line) and the L train (the grey line).

The 4-5-6 line stops at 42nd Street-Grand Central.

To get to the A or C, take the S shuttletrain (color grey) west across 42nd Street, from Grand Central (East 42nd) to Times Square (West 42nd). Then walk through the tunnel that connects the 42nd Street-Times Square Station to the 42nd Street-Port Authority Bus Terminal Station. At the 42nd Street-Port Authority Station, get the Downtown A or C.

Take the Downtown 6 train to Bleeker Street. On the Downtown 6 only (not the Uptown 6) you can transfer from the Bleeker Street Station to the Broadway-Lafayette Station.

Get the Downtown B, D or F trains at the Broadway-Lafayette Station.

Take the S shuttle train (color grey) west across 42nd Street, from Grand Central (East 42nd) to Times Square (West 42nd). At the Times Square Station, transfer to the Downtown N, R or Q trains.

Take the S shuttle train (color grey) west across 42nd Street, from Grand Central (East 42nd) to Times Square (West 42nd). Get the Downtown 2 or 3 at the 42nd Street-Times Square Station.

Take the Downtown 4-5-6 to 14th Street-Union Square. It's preferable to take the 4 or 5, not the 6, as the 4 and 5 are express trains and it's only one stop from Grand Central to 14th Street. It's four stops on the 6local train.

Get the Brooklyn-bound L train at 14th Street-Union Square.