Loretta, I love real estate, but I'm not a realtor. Why spend 1.2 M in the area? You can get many fabulous houses in the area for less than that. The time is still very good in the real estate market in Naples.
How about a great 5 bedroom, 4.5 bath, 3 car garage house in Naples? It is about 4,340 square feet, spacious, looks like it is there in conjunction with the Valencia Golf and Country Club for a fairly small fee (under $300 a month). Get those kids involved with golf! I clicked on the schools tab on Trulia and it said the schools are good to excellent. It has a small lake view. It's located at 1750 Sarazen Place, Naples, FL 34120. Of course, I am not sure if there are lots of kids in the area but you would have to check it out. Remember, I think realtors have to be totally neutral when it comes to recommending schools and "neighborhoods with children", but I think they can tell you about their own experience if they came to Naples with children and a family.
Of course, this house could be sold already, but this is just an example of a great house for $425,000 which is no where near 1.2 million dollars. Go to realtor.com and look for free, then contact a realtor that will work for you to show you what you want to see or to find you what you want. When I found our house, I also thought it was important to find a realtor who you can be honest with, someone you can say, "sorry this one isn't for me" , where they won't get insulted. You DO have to tell your realtor along the way what you LIKE about houses, too. I also think it's important to remember that you can click on the "crime" tab of each individual listing on Trulia to see what the crime rate looks like or contact the local police in the area and ask about your particular neighborhood.
My parents love their condo on Citrus Lake in North Naples, but theirs is only a 2 bedroom, 2 bath modest condo. There are some stand-alone condos around that lake, but I'm not sure if they would be large enough for your family or luxurious for what you have in mind. It's about 5 miles from any beach, including Doctor's Pass and Vanderbilt Beach. It's a gated community with two pools, fishing off the pier, a hottub, basketball court, shuffleboard, and tennis courts, and a walking path around the lake.
P.S. Instead of spending 1.2 million on a house, take some of the money you save to do fun things with the kids. Maybe buy a boat, take them swimming and fishing, take them to the excellent symphony in Naples and to do other cultural things in the area, church events, plays, boating excursions, fun restaurants. They won't remember a $1.2 million dollar house, but they'll remember that you spent time with them and their friends doing things together as a family.