POTENTIAL CONFLICT WITH SAME DIRECTION TFC.

Narrative

I HAVE BEEN FLYING CAPT ABOUT 1 MONTH DOMESTICALLY AFTER 2 YRS OF INTL FLYING. I'VE BEEN SURPRISED BY THE NUMBER OF IMMEDIATE CLBS; DSCNTS; TURNS THAT I'VE BEEN GIVEN BY ATC IN THE LAST MONTH. ON THIS SAME TRIP WE ALSO GOT AN RA FROM A LIGHT SEL FLYING THROUGH THE BNA ILS COURSE 20L. HOWEVER; WE HAD GOOD VISUAL CONTACT ON HIM PRIOR TO THE RA AND AVOIDED HIM BY DEVIATING COURSE BEHIND THE INTRUDER. MY GUT QUESTION IS;' IS OUR POSITIVE CTL ATC SYS COMING UNGLUED?' CLBING 250 KTS TO 13000 FT ON DALLAS 8; DFW DEP. TA NOTED ABOVE AND AHEAD R. LOOKED ABOVE AND FORWARD BUT NO CONTACT. RA NOTED; WE REDUCED CLB; LEVELING ABOUT 9200. ATC THEN CALLED TELLING US TO MAINTAIN 9000 FT AS WE WERE BEING OVERTAKEN FROM BEHIND. DSNDED TO 9000 AND OBTAINED VISUAL CONTACT AHEAD AND TO THE R IN A R HAND TURN. WE HAD DEPARTED RWY 35; MAINTAINING 360 DEGS MUCH LONGER THAN NORMAL. THEN TURNED TO HDG 110 DEGS TO INTERCEPT 067 DEG DFW WITH 8000 FT RESTRICTION. LATER CLRED ABOVE 10000. INTRUDER OVERTOOK US FROM BEHIND. I DO NOT KNOW WHY INTRUDER WAS AT THAT ALT OR SPD NEAR DFW.