MD88 EXPERIENCES NMAC AS A RESULT OF HEARBACK READBACK ERROR ON CLEARED ALT.

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WE DEPARTED OFF OF RWY 8 IN PNS. INITIALLY CLRED TO 'FLY RWY HDG; MAINTAIN 3000 FT.' UPON CLBOUT; TWR INSTRUCTED US TO CONTACT DEP. UPON CONTACTING DEP; WE WERE TOLD 'CLB AND MAINTAIN 10000 FT.' CAPT AND I BOTH ACKNOWLEDGED THE 10000 FT ASSIGNED ALT. WE WERE CLBING THROUGH 3600 FT MSL WHEN DEP ASKED US 'WHERE ARE WE GOING?' I INFORMED THE CTLR THAT WE HAD BEEN PREVIOUSLY CLRED TO 10000 FT. THE CTLR THEN TOLD US TO 'TURN L TO A HDG OF 360; MAINTAIN 4000 FT.' I PLACED 4000 FT IN THE ALT ALERTER. WE WERE CLBING RATHER QUICKLY AT THAT POINT; SO THE CAPT DECIDED TO DISCONNECT THE AUTOPLT SYS TO TRY AND MAKE THE 4000 FT LEVELOFF. WE WERE ALSO TURNING N AT THAT POINT. WE FLEW THROUGH THE 4000 FT ALT AND AT THAT POINT WE RECEIVED AN RA FROM TCAS. INITIALLY; THE RA WAS FOR US TO DSND. THE CAPT WAS 'HAND FLYING' THE ACFT AT THE POINT AND HAD INITIATED A DSCNT. AT THE TIME; THE COCKPIT WAS 'ABUZZ' WITH AUDIO ALERTS FROM THE ACFT SUCH AS 'ALT ALERT AND DSND!' UPON DSCNT; THE CAPT AND I BOTH NOTICED ON THE TCAS SCREEN THAT THERE WAS AN ACFT BELOW US THAT APPEARED TO BE CLBING IN OUR DIRECTION. INITIALLY; THE ACFT WAS 800 FT BELOW US. WE FOLLOWED THE RA DSCNT; UNTIL WE BOTH NOTICED THAT THE CONFLICTING ACFT WAS DIRECTLY BELOW US AND STILL CLBING. AT THAT POINT; THE ACFT WAS 400 FT BELOW US. I POINTED OUT TO THE CAPT THAT WE NEEDED TO CLB. HE INITIATED A CLB AT THAT TIME. A FEW MOMENTS LATER; THE TCAS SYS TOLD TO 'CLB.' I LOOKED AT THE TCAS SCREEN; IT INDICATED THAT WE WERE ONLY 200 FT APART IN ALT AND NEARLY 'ON TOP' OF THE CONFLICTING ACFT. WE CLBED TO 5000 FT AND CONTINUED ON. WE RPTED TO ATC THAT WE HAD RECEIVED AN RA.