Hello, and welcome to Just Answer!
I think you will find the tips at the following websites helpful:http://www.mayflower.com/moving/full-service-movers/moving-tips/moving-checklist.htmhttp://www.upack.com/moving-resources/checklist.asp
Because you have already received many tips from your moving company and friends, there might be information you already know, and also new information, at these sites.
This is what I would suggest you take with you in your suitcases: personal items, toiletries, medications, landline and/or cordless phone (with adapter for U.S. outlets), answering machine, if you have one, laptop computer, favorite toys and bedding for the children, bedding for you, clothing that will be suitable for warm weather and cooler weather; also, rain gear like raincoats, boots(?) and umbrellas for all. I live in South Florida, and you didn't mention what part of Florida you would be moving to, but the northern part of the State is typically cooler than the southern part. Right now, most of Florida, but especially South Florida, is still experiencing hot, humid weather, with a chance of rain, every day. If you're moving to a different part of the state, it might begin getting a little cooler over the first few weeks you are here, hence the need for the appropriate types of clothing. If the children are of school age, they will need school clothes and play clothes, of course, and you can bring anything you have now, and/or buy their school supplies when you arrive and get lists from their teachers. If they're not in school yet, of course, you don't need to include this. But, do include coloring books, crayons, reading books and favorite toys/stuffed animals.
You will need a minimum number of pots and pans for cooking, dishes (paper plates and plastic cups are great, during the transition, but some meals will require 'real' dishes) which can be bought rather inexpensively, in stores like Target, Walmart or Kmart, when you arrive, unless you are moving into a furnished place which may have some of these items already. Try to buy as many new things as you can, when you arrive, and pack everything else. For example, if your place is unfurnished, you will need window treatments like blinds, shades, curtains, etc. If you have windows which let in the sun at certain times of day, you might want to place room darkening shades on them to cut down on the intrusion of heat, and saving on your air conditioning usage.
You can buy an inexpensive 'bridge' table and folding chairs to eat your meals and use as a 'desk', until your furniture arrives.
It might be a good idea to think of each individual room and what you are used to having/using, now, on a daily basis, and then include these items that are convenient to pack, in your suitcases or boxes you will be bringing with you on the plane, and the larger items will be included in the shipment that will arrive first. Go room to room, now, and make your lists of things to 'keep', 'throw out', 'give away' (rummage), 'pack in suitcases' (for each family member), 'pack in boxes coming on the plane(?)', and 'first shipment', 'later shipment'.
I made the move to Florida many years ago, with my two young children, but from another State, and we drove, but the same logistics re: packing the car, suitcases, and what we needed until the moving van arrived, applied. One difference was, we stayed with my parents until the moving van arrived and we were set up in our new place, so all the comforts of home were there for us. I don't know if your situation will be the same, i.e., staying with friends or relatives, at first, but hopefully the information here, will help you.
I wish you much good luck with your move, and hope all will go smoothly!
I hope you have found this answer helpful; if so, please click 'ACCEPT' and your Positive Feedback will be greatly appreciated. If you need additional information or have further questions, please click 'Reply'. Thanks for the opportunity to be of help.