According to Florida law, http://law.onecle.com/florida/motor-vehicles/316.123.html, when you are approaching a four way stop, the first person to come to a complete stop is given the right to pull through the intersection. If more than one person stops at the same time, the person to the right has the right-of-way and may proceed through the intersection. This is the rule for all of Florida, including Gainesville.
However, as an experienced accident lawyer, I caution you to make sure you are aware of what the other drivers are doing. Make sure they see you and know you are entering the intersection before them. When in doubt, stop, wait, and be safe.
For more information:
What are the Proper Intersection Rules in Florida?
What do I do at a yield sign?
Look both ways before you enter the intersection