What do you like to do on trips? Restaurants, museums, tourist attractions, explore out-of-the-way neighborhoods? A lot would depend on what you, personally, like to do.
As a resident of Brooklyn, though, I would highly recommend a trip to my borough. A walk across the Brooklyn Bridge from Manhattan (City Hall) is an amazing way to see the City. From the bridge, you can walk to the historic Brooklyn Heights district, go to the Brooklyn Heights Promenade, then go a bit further south to Cobble Hill and Carroll Gardens where there are dozens of restaurants and bars, all of which are just as good (and in some cases better than) Manhattan restaurants and bars, but half the price. There are half a dozen subway lines that can take you back to Manhattan if you're too tired to walk back to the City. This area is extremely safe and accessible (despite what Manhattanites will tell you).
Also, check out timeoutny.com for listings (art, music, dance, etc.) and newyorkmetro.com (New York Magazine's website) for restaurant recommendations.
By all means go to the Statue of Liberty and the Empire State Building, but be a little adventurous, too. New York has a lot to offer if you just dig below the surface a little.