Reading through some of the other comments it seems you are open to several options. I love JC and lived there for 6 years, recently moved but still check in here because I love the city. If you want to be in JC then I definitely recommend Hamilton Park or Paulus Hook. You'll be hardpressed to find personal outdoor space though. I lived in the city with a dog and it was fine, city is very dog friendly.

I see you are from a shore town, if you like the beach I would echo some other folks comments about Monmouth county. I recently bought a house in Red Bank, NJ and would highly recommend. It's more urban than what you are probably used to but not Jersey City Urban. It's a 10-15 minute drive to the beach and there is an awesome live music scene and walkable downtown. Every bar has live music Weds-Sun night all year long. Several other shore towns within 15-30 minutes have thier own bars, breweries, and downtown areas. If you like fishing the Jersey Shore has some of the best in the country. The area gets very lively during the summer. It's a 45 minute drive to Jersey City or you can take the train to Newark and catch the PATH. You can get a car and won't have to pay extra for parking as well. All in all it's really about your priorities. Red Bank seems to have alot of Ex Pats as I met several English folks at a local pub during the USA/England world cup match.