Beautiful house. Keep the palms. What spoils it is the wrought iron work and the supporting corner post, inconsistent with the Nouveau Florida design of the house. if you must have an enclosure for children in fron, there is a better way of doing it. I feel that what should be there is a stone patio surronded bu plantings for privacy (and child safety). That will look nice from the interior looking out, as well. Would use a fountain and probably some statuary. In fron of the palm tree area, would uses some curved annual flower beds in brilliant colors.

I do not object to the iron fencing, but absolutely agree with nevadan that you should keep the palms. Hiring a local landscaping professional could be of great benefit to you. A landscape designer could draw a plan for you, using local plants with which you may not be familiar. With that plan, you could implement it yourself or hire someone else to do it.

I love the palms! Try adding some colorful, low maintenance plants, like small flowers.

Lovely home, and gorgeous tall palms are a nice balance to the dramatic height of your roof line. It looks like there is quite a bit of space to work with. As stated by others, your best bet is to hire a landscaping professional. If you are unsure, take some time to visit and walk through some nice nurseries in your area. If you find a nursery you like, ask for their recommendations for landscaper / designer.