I replaced my damaged tailgate with one from a junk yard that I repaired and painted but I used the 2 faced tape from Pepboys and the wide black plastic trim fell off after a week and a half what can I use that will hold.

 

3m tape. I have found it at most parts stores. I think it's a 1/2" wide and thicker than normal tape, it's foam like. It bonds parts like that extremely well.
Should be in a red package and the tape is gray/black.

 

Second the 3M tape. As long as you clean the surfaces with alcohol the stuff will stick forever.

Testimonial: I used it to attach a plastic switch box to a painted steel tank that is always about 210°F. Didn't think there was any way it would hold, four years later it's still rock solid.