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The Cure, 1986 in Philadelphia. I was a high school senior. To be honest, I don't remember much. The second (and third, fourth and fifth) was the Grateful Dead, on that same tour with Thomas Dolby that Masswich mentioned.
Aerosmith (with Fuel as the opening act) at Alpine Valley Music Theater in East Troy, WI in July 2001.
I also saw folk artist Leo Kottke at the Woodstock Opera House sometime before then, but I consider Aerosmith to be my first true concert, as it was a large venue and a rock show.
I have been to probably 60-75 concerts or festivals since then, including seeing many of my favorite bands (several more than once), with Aerosmith far ahead in the lead with 10 shows. I consider myself pretty fortunate to have seen so many great acts and great live performances.