nybbler's right. NYC's bike infrastructure is light years ahead of anywhere in NJ. NJ has some nice parks where you can ride, but I get bored really quickly doing 1-mile laps around county park X or town park Y. In NYC (and most civilized cities anymore) there are bike lanes, paths, etc. that you can use to connect to other parks and points of interest.
Look at the NYC bike map for some ideas.
NYC DOT - Bicycle Maps. When I got my bike I would just pick a destination and then look at the bike map or google to find a good, safe, and/or scenic route there. It was a great way to familiarize myself with a lot of the outer borough neighborhoods I didn't know much about.
Perhaps the most popular ride in the region outside the city is to go over the GWB and ride north on the Jersey side along Route 9W up into Piermont and Nyack, NY. It's a long ride with some good hills, and plenty of chances to turn around if you get worn out. There are a lot of lycra-clad "pros" out there, but also a lot of more casual riders out for fun. There are also some bike shops, delis and other stopping points that are filled with cyclists. On a good day, people will be chatty and you may make some new riding buddies. Just head for the bridge on a weekend morning and follow the bikes north.
Another fun one is to cross over the Brooklyn Bridge, meander your way down to Prospect Park, follow Ocean Pkwy down to Coney Island, and reward yourself with a hot dog.
If you're looking to make a good day trip out of it, go a little farther away and do some road riding on the North Fork. If you're into dirt riding, take a train or drive down to New Brunswick and ride along the Delaware & Raritan Canal towpath. I've done the section from NB to Princeton & back a few times, but the trail continues to Trenton and then up the Delaware to Frenchtown. In fact there's another towpath trail on the PA side of the Delaware so you could do a loop from Frenchtown down to Lambertville, cross to New Hope and ride back up the PA side to Erwinna and back over to Frenchtown.
Lastly, look into some of the cycling groups like Bike New York or the Morris County Freewheelers. These groups put on a few rides each season that you can participate in, or steal the cue sheets for and ride on your own one weekend. I have to recommend MCFW's Revolutionary Ramble in June. My favorite organized ride. Good crowd, well managed, and a fun, scenic ride. You don't need to be a "pro" to do these rides, either. Lots of people on fat wheels and wearing shorts and sneakers, just out for the fun too!