Yes, you have a legal duty to support your child. If you don't then the taxpayers have to and we don't want to. The same judge who orders child support can order visitation. Failure of the mother to obey the visitation order is a contempt of court and can be punished, theoretically, by the judge upon your application.
The only way you are absolved of your responsibility for the child you produced is for the mother to marry and for the new husband to adopt the child. Otherwise, you are responsible unitl the child is 18. Furthermore, all the 50 states have a compact between them. If you move to another state, that other state will enforce the Florida child support order. Florida will track you down through your social security number. If you don't pay the child support your wages will be garnished, your drivers license suspended, and eventually you will be jailed for contempt of court. Your wife does not have to employ and attorney to enforce payment of child support. The state employs nasty women lawyers to chase after deadbeat dads without mercy. Sorry, but those are the facts.
Unfortunately terminating your parental RIGHTS will not do anything to terminate your parental RESPONSIBILITIES (such as child support). While some other states have permitted the termination of a parent's obligation to support a child upon the termination of their rights (or voluntary surrender of them, etc) Florida has taken a clear stance that doing so is against public policy and your "responsibilities" will only be terminated if someone else "steps in" to your shoes and adopts the child in your stead. The most common of these is via a consensual step-parent adoption such that your rights/responsibilities are terminated while another gentleman's are started - and even then you may still be on the hook for whatever child support was owed up until the date of the adoption (though it is one of the rare exceptions wherein the parties are permitted to waive/release the past due child support in exchange for the consent to the adoption).
Accordingly, your obligation to pay child support WILL be treated entirely independently from whether or not you elect to file an action to seek a "time sharing" schedule or "parenting plan" with the child (Florida no longer uses terms such as "custody" or "visitation").
If you have any questions about your case specifically, what your rights are, how best to proceed, etc., I would recommend you contact a Family Law attorney in your area for a consultation to learn more and can better protect yourself.
Child support Child custody Child support and custody Finances and child support Determining child support payments Ending child support Child support enforcement Enforcing child support by contempt Mother's rights in child custody Parental rights in child custody Social security Government law Lawsuits and disputes Family law Parental responsibility Contempt of court