Most counties allow property owners to clear overgrowth without a permit. Depending on the county, overgrowth is generally defined as all brush and small trees up to three or four inches in diameter. Palmettos and other invasive species do not require permits.
Any large trees will generally require permits to take down unless zoned agriculture. These should be removed by a licensed Tree Service company. ShadowCat can refer you to legitimate Tree Service companies once you are at the point of removing big trees.