FLC CLBING TO FL230 IN DC9 AND IN CONTACT WITH ZMA RECEIVE TA ABOUT VFR TFC AT 13500 FT. THE CTLR STARTS A VECTOR AWAY FROM THE TFC. THE CREW THEN RECEIVES A TCASII RA AND TAKES EVASIVE CLB ACTION. APPARENTLY ANOTHER ACFT SETS OFF THE TCASII AND THE CREW RECEIVE A DSND COMMAND.

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FO FLYING; CLBING THROUGH APPROX 12500 FT; ZMA (124.1) ADVISED US OF VFR TFC AT 13500 FT. WE WERE CLBING TO FL230. TFC WAS OBSERVED ON TCASII BTWN 12 - 1 O'CLOCK AT APPROX 6-8 MI AND CLOSING. I TOLD THE CTLR THAT WE WOULD OR COULD LEVEL OFF AT 13000 FT. HE ISSUED US A L TURN FROM APPROX A 340 DEG HDG TO A 270 DEG HDG. AS WE WERE ESTABLISHING THE TURN WE RECEIVED A TCASII RA TO CLB. THE FO ESTABLISHED A POSITIVE CLB ATTITUDE; RA REQUIREMENT WAS BTWN 1500-1800 FPM CLB. ALMOST IMMEDIATELY AFTER OUR CLB WAS ESTABLISHED; WE RECEIVED AN RA TO DSND. WE CONTINUED OUR CLBING L TURN. AS WE WERE IN OUR CLBING L TURN; I SELECTED THE 6 MI RANGE ON MY TCASII INDICTOR FROM THE 12 MI RANGE. THE OTHER ACFT WAS WITHIN THE SMALL INNER RING. AFTER A COUPLE OF CALLS TO ZMA; THE ONLY COMMENT HE HAD ABOUT THE TFC WAS THAT ORL APCH HAD OR WAS HANDLING THAT TFC. I MAY HAVE CAUSED THE CTLR A MOMENTARY DELAY IN ISSUING INSTRUCTIONS BY SAYING THAT WE WOULD OR COULD LEVEL OFF AT 13000 FT. I SHOULD HAVE TAKEN CTL OF THE ACFT FROM THE FO AND LEVELED OFF AT 13000 FT AND ADVISED THE CTLR OF MY ACTION AND ASKED FOR FURTHER INSTRUCTION. I DO NOT UNDERSTAND WHY THE CTLR PUT US IN THE POS TO CLB THROUGH THE ALT OF THE OTHER ACFT; CAUSING US TO HAVE TO MAKE DECISIONS ON HOW TO AVOID A MIDAIR COLLISION.