In the state of Florida, you'll need to be licensed if you perform any of the following real estate activities for someone else for compensation:
The reason for licensing, quite simply, is to protect the general public. There are some special situations that are exceptions, but the type of business you're contemplating doesn't sound like one of them.
Property management definitely involves renting and leasing other people's properties for a fee. For further explanation of what does and does not require a real estate license, refer to the Florida Department of Business and Professional Regulation (DBPR) webpage cited below:
You'll also find the phone number for the DBPR there as well. I hope that helps.