IFR corporate aircraft landing APF experienced TCAS RA with ATC reported traffic; traffic not sighted until RA maneuver.

Narrative

A Beechjet was in the terminal/approach phase of a FAR Part 91 passenger flight. Prevailing weather conditions were day; VMC. We were being vectored by Ft. Myers Approach for a visual to Runway 23 at APF. VFR traffic was reported by FT. Meyers Approach at our 12 o'clock position and 500 FT above our assigned altitude; to which the pilot not flying stated 'Looking for traffic.' Soon after; we received a TA from TCAS and still were not able to acquire the target visually. We then received an RA to descend from TCAS. I immediately disengaged the autopilot and initiated a descent. At this point; the pilot not flying advised Approach that we were responding to an RA. Approach then advised us of additional traffic 1;000 FT below us as we simultaneously received another TA from TCAS. The pilot not flying requested a vector to avoid traffic as we were unable to acquire either target visually. Approach gave us a 180 degree heading and while in the turn; we were able to establish and maintain visual separation with the aircraft at the lower altitude. The remainder of the approach and landing proceeded without incident.