As a life long NYC area resident & former realtor, I hate to bring bad news but you have several challenges:
1. Apartment buildings in the NYC area are not pet friendly. Very few rental buildings allow pets because the additional pet insurance for landlords is very expensive and landlords refuse to pay for it. Therefore, it's much easier and cheaper for landlords to put up "No Pets" signs than pay for pet insurance. What you can do is pay for your own pet insurance and get a Good Canine certificate to show landlords.
2. Depending on the community, a 2 bedroom 2 bath rental ranges from $2,500. - $5,000./month. The average is $3 - 4,000./month. There are very few rentals with 2 bedrooms & 2 baths. The majority are 2 bedrooms, 1 bathroom or 3 bedrooms, 1 - 2 bathrooms. Most rentals either have parking on site or parking within a short walking distance. Either way, parking is an additional fee - it's never free. Also do not expect any amenities. Only the luxury rental buildings have amenities and a 2 bedroom 1 bath = $4,000./month.
3. As for communities - if you are renting, pay attention to the commuter railroad stops. Many rental buildings are within a short walk to the commuter railroad. The railroad lines - PATH, MetroNorth, Long Island Railroad & Passaic Hudson. Look at the express stops & local stops & check out rental buildings along the stops. This is what the management companies use to advertise - the convenience of the walk to the train station. In most cases it's a 30 min. Or less train ride to Manhattan.
4. One tip/advice - I sincerely hope your income is at least $150,000./year. Below $150,000. means you will barely survive. You won't have anything extra to enjoy life - just enough to pay your bills. Above $150,000. Means you will have money left after your bills to enjoy life.