We have been living here for over 10 years now.
Rent (unfurnished) is about $1200-1500 for a 2-3 bedroom ($800-1000 for a 1 bedroom) at a decent place where your life is not in imminent danger after 8PM. They do run credit checks on you and ask for references, so not sure how difficult it is to get a place without those. Perhaps your school can arrange it. Security deposit is about $500 most places. This assuming you sign a 12 month lease. Month to month is possible for significantly (15% or so) more. Each city has bad and good areas. Things on the East side of I-95 are often more dangerous but it is hard to pinpoint where not to go.
Utilities:
Water, Electricity is about $100-200 depending on the season. Landline phone about $30, cell about $50-150 depending on what you want and how many lines. Cable TV $60, internet $60. You can get combo on phone, cable, internet and save significantly. Better skip it all and get a calling card or just have internet and use skype.
Car insurance about $60-100 a month especially if you have no US driver history. You can get a beaten, but working used car for $3000 that you may be able to sell for the same when you are done. You will not be able to lease and long term rental is a rip off imho. But not having a car is not an option. Not like in Europe. Forget about bicycles too because of the heat, and the drivers. Very unsafe. Fuel is cheaper than back home. Regular is $2.60/gallon at the moment.
Me and my wife use about $150 a week for regular groceries. That includes supplies and food. We do cook each week, not to save, but to eat healthy. One time eat out can be from $60 at a regular chain restaurant like Chilli's for 2 persons. That is a full menu with several beers/wine. You can get specials for less, and get away with $20-30 for two persons but no alcohol. Fast food runs you about $15 for two persons. There are some relatively healthy fast food places like Pollo Tropical. It is not all hamburgers.
I would suggest you to consider coming here without the family and stay focused, fly each day if you can. Get as much studying done as possible before you get here. You can finish up a commercial multi very fast if you are focused, and half decent. I used to instruct, and in my former school some capable folks went zero to MEI in 12 months and 270-300 hours. That is PPL, IFR, COMM single, COMM multi, CFI, CFI I, MEI. So you doing comm only should be a matter of weeks even if you have to build hours. So why spend all that money on getting the family over here? Florida is not an awesome vacation place if that is what you have on your mind. Beach is so-so, and the rest is a swamp. It really does not compare to the Mediterranean. In addition I would be very hesitant bringing kids that young. Consider the extreme medical expenses that may arise. A simple visit to the doctor is $100, once you step into a hospital it gets really expensive. Food poisoning is fairly frequent here in part due to the heat.
Good luck!