Jellybean is essentially correct.
First of all, Brooklyn is IN NYC. You mean that you want to go into Manhattan.
Second of all, where you are staying there are lots and lots of subways into Manhattan.
Jellybean recommended the R train to Times Square. Well, yes, I suppose you could do that. But the R takes forever. It stops everywhere.
A better bet would be the number 2 train to Times Square. That also stops right near your hotel. You just have to cross Adams Street (carefully, it's a busy street!) Turn left and go about a block. Turn right and there you will see the subway station with many different subway lines. (That station has the 2,3, 4, 5, R and M lines running through it.) Get a subway map from the clerk in the booth and figure out where else you want to go by subway.
That's a very well served area by subway, though it's an ugly location right by the hotel. Get out of the immediate area by walking past where the subway is and keep going. Eventually you will end up in Brooklyn Heights. Turn to the right after a few blocks and find Montegue Street. Walk to the end and you'll get to the Brooklyn Height Promenade and a fantastic view of the Manhattan skyline!
Walk north a little more and you'll get to a lovely section of Brooklyn heights with 18th Century houses (the streets all have fruit names: Cranberry, pineapple, etc.) Wander around Brooklyn Height a little, you'll get a much better sense of where you are then if you just go to the hotel and from there to Manhattan!
You might in the other direction towards Metro Tech to catch a subway farther out in Brooklyn to Park Slope, one of the trendiest neighborhoods in Brooklyn. There are tons of fantastic restaurants and bars, lots of cafes and boutiques.
Many people find Brooklyn to be a wonderful place to live, and many European tourists have now started to come to Brooklyn when they visit NYC. Give it a shot!