I agree that experience with a variety of weather is useful, but also think that it is helpful to start building your skills in generally benign conditions. Then transition to more challenging conditions. If you agree, then starting in May or late Sept/early Oct are, IMO, the best fits for NJ.
Typically the current weather pattern of rain followed by cold/blustery...repeat...diminishes by late April. It would give you May and early June to fly before the worst of the summer convection season. And even then, if you can train in the early AM, the weather should be good for consistent training into the summer.
Starting in the fall also usually provides good weather for several weeks. But, IMO, training in the winter is more challenging than in the summer. Cold, snow, and ice seem to have a more lasting effect on the schedule than convection and storms.