I live in Hoboken and used to live in the waterfront area of Jersey City and although I love each areas, if your budget is $2000-$2500 and you want a "modern place" that isn't very urban, I wouldn't suggest either for you. Both Hoboken and Jersey city are very urban and densely populated and that price range is on the very low end of what it would take to rent here--guaranteeing a 3-5th floor walkup, no parking garage, and definitely no gym in the building. Elementary schools are okay, after that it gets slightly more questionable. There are a small handful of places on River Road in North Bergen and Edgewater that would be an easy commute to Midtown (via ferry or bus) that are modern, have a gym, close to shopping, and are in your price range.
Half Moon Harbour Apartments-North-bergen | Property Floorplans I actually really like that area--beautiful trail/sidewalk area right on the river with a view of the city that is perfect for walks or jogs.
Since you are open to up to a 1 hour commute, there are tons of other places you can research that are on the train line that would have slightly cheaper 2 bedroom rentals: Montclair, Maplewood, Ridgewood, Westwood, Paramus. The first 4 are smaller but very nice towns with a sort-of "high street" or main street with local shops. The last, Paramus, is much larger but has tons of shopping and a lot of apartment choices/house rentals for a lower price.
I like the other poster's ideas of just renting for a year until you get a better feel for NJ. Welcome! Feel free to ask any other questions.