Yes, most states, including Florida, will use the criminal records from other states.  However, sometimes their use will be limited.  For general punishment evidence, almost all of a defendant's prior criminal conviction history will be admitted and used.  Prior convictions that are used for specialized enhancements are more limited.  For example, driving while intoxicated offenses are frequently enhanced if a defendant has a prior conviction for driving while intoxicated.  States will generally require that the nature of the offense for which the defendant was convicted in another state, be substantially similar to the driving while intoxicated laws in the charging state.  If they are not substantially the same, they may be disqualified to be used as enhancements, but they could still be admissible as general punishment evidence.