A DEP B727 ACR HAS A PROB WITH THE DEP PROC OUT OF FLINT; MI; ENCOUNTERING INBOUNDS FROM THE S.

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PRE-DEP CLRNC WAS 'ACFT CALL SIGN; CLRED TO DTW; RADAR VECTORS; POLAR ONE; MAINTAIN 3000 FT; EXPECT 6000 FT IN 10 MINS; DEP CTL FREQ 118.8; SQUAWK XXXX.' DEPARTED FNT (FLINT; MI) ARPT RWY 27; CLRED RWY HEADING BY TWR. UPON HDOF TO DEP CTL WE WERE INSTRUCTED TO FLY RWY HEADING CLB TO 6000 FT FOR RADAR VECTORS TO (HERE COMES THE CONFUSION) SOME INTXN (I BELIEVED IT WAS AN INTXN AT FIRST) THE NAME I DID NOT CATCH BECAUSE IT SOUNDED STRANGE (NOT ONE THAT I WAS EXPECTING TO HEAR). CAPT READ BACK CLB TO 6000 FT FLY RWY HEADING FOR WHAT HE BELIEVED TO BE THE CLRNC. NO CORRECTIONS WERE MADE BY DEP CTL. WHILE I WAS FLYING THE ACFT THROUGH CLEAN UP CONFIGNS AND NOISE ABATEMENT PWR SETTINGS THE CAPT ADVISED ME I WAS FLYING THROUGH THE VOR RADIAL THAT I WAS ON RWY HEADING TO JOIN TO GO ON TO THE POLAR ONE ARR AS DEP CTL WANTED. I TURNED L AND JOINED WHILE CLBING AND SETTING PWR FOR A SHORT CLB TO 6000 FT. (THIS ALL TOOK PLACE BEFORE I COULD RE-LOOK AT MY ARR PLATE TO CATCH THE FIX DEP CTL WAS VECTORING US TO; SO; I BELIEVED IT WAS JOINING A RADIAL/AIRWAY INSTRUCTION THAT I DIDN'T CATCH -- NOT THE NAME OF A FIX.) AT ABOUT 4000 FT MSL ATC ADVISED US TO MAINTAIN OUR PRESENT ALT AND TURN TO 270 DEGS THEN 330 DEGS (R TURN). DURING THE LEVELOFF; PWR REDUCTION AND R TURN ANOTHER ACFT APPEARED ON TCASII ABOUT 3 1/2 TO 4 1/2 MI AWAY SEVERAL HUNDRED FT ABOVE OUR ALT. THE OTHER PLT ADVISED US IN SIGHT TO ATC. OUR CAPT HAD HIM IN SIGHT ALSO. NO TCASII WARNING WERE SOUNDED.